DAMAGE BOOK 00 a: OU 160464 >m A CONCISE GEOMETRY BY CLKMENT V. IHJRELL, M.A. SJ.NIOI' MAIIIFMAIHAI MAS IRK, \VTN< 111 SI PR COT I Kf>P LONDON G. BELL AND SONS LTD. 192 T First Publhhcd . . January IQZI Reprinted August IQH PREFACE TITE primary object of this text-book is to supply a large number of easy examples, numerical and theoretical, and as varied in character as possible, in 'the belief that the educational value of the subject lies far more in the power to apply the fundamentoKfrte of geometry, and reason from thorn, than in the ability to reprocfoice proofs of these facts. This collection has grown out of a get* of privately printed geometrical exercises which has been in U&Q, for many years at Winchester College : the author is indebted to many friends for additions to it, and to the following authorities for permission to use questions taken from examination papgrs : The Controller of His Majesty's Stationery Office ; The Syncujj of the Cambridge University Press ; and the Oxford and Cambric^ Joint Board. Riders are arranged in exercises corresponding to groups of theorems, and Constructions are treated similarly. There is also a set of fifty Revision Papers. Answers are given to all numerical questions, except where no intermediate work is necessary. Harder questions and papers are marked with an asterisk. The book is arranged as follows : I. Riders, Numerical and Theoretical. II. Practical Geometry ; Construction Exercises. III. Proofs of Theorems. IV. Proofs of Constructions. The Proofs of Theorems and Constructions are collected together instead of being dispersed through the book in order to assist revision by arranging the subject-matter in a compact form. When learning or revising proofs of theorems and constructions, it is most important the student should draw his own rough figure. For this reason, no attempt has been made to arrange the whole of the VI PREFACE proof of a theorem on the same page as the figure corresponding to it. The order and method of proof is arranged to suit those who are revising for examination purposes, and is not intended to bo that used in a first course. It is now generally agreed that proofs by superposition of congruence tests and proofs of the fundamental angle property of parallel lines should be omitted in the preliminary course, but that these facts should be assumed without formal proof and utilised for simple applications, the former being treated by some such method as that noted on page 14, and the latter by rotation or the set-square method of drawing parallel lines. This broadens "the basis of the geometrical work and enables the early exercises to be of a more interesting nature. The arrangement of riders in one group and practical work in another is made for convenience of reference. Naturally both groups will be in use simultaneously ; but the course should open with the exercise on the use of instruments in the practical geometry section. No attempt has been made to include in the text the usual - preliminary oral instruction which deals with the fundamental concepts of angles, lines, planes, surfaces, solids, and requires illustration with simple models. The examples start with methods of measurement and general use of instruments, which is the earliest stage at which a book is really any use for class work. The object throughout has been to arrange the book to suit the student rather than the teacher, and " talk " is therefore cut down to the minimum. It is the nature of the examples which has been the chief consideration, and if this part of the book receives approval, the author will consider his object has been attained. Valuable assistance has been given by Mr. A. E. BROOMFIBLD, without whose advice, interest, and encouragement the work could scarcely have been carried out. C. V. D. August, 1920. CONTENT'S R1DEHS ON BOOK I PAUH ANGLES AT A POINT ...... 1 ANGLES AND PARALLEL LINES ..... 5 ANGLES OF A TRIANGLE, ETC. . .9 ISOSCELES TRIANGLE?, CONGRUENT Tin ANGLES ( FIRST SECTION) . 15 CONGRUENT TRIANGLES (SECOND SECTION), ETC. . . .21 RIDERS ON BOOK II AREAS . ... ... 25 PYTHAGORAS' THEOREM .... 37 EXTENSION OF PYTHAGORAS' THEOREM . . 43 SEGMENTS OF A STRAIGHT LINE ..... 46 iNEtjUALI'IKK . 48 INTERCEPT THEOREM .... 51 RIDERS ON BOOK 111 SYMMETRICAL PROPERTIES OF A CIRCLE . .56 ANGLE PROPERTIES OF A CIRCLE ... .60 ANGLE PROPERTIES OF TANGENTS ... 67 PROPERTIES OF EQUAL ARCS . . . . . .71 LENGTHS OF TANGENTS, CONTACT OF CIRCLES . . .76 CONVERSE PROPERTIES ... 82 MENSURATION ... . .86 Loci ......... 93 ClRCUMOlRCLE ....... 99 iN-ClRCLE, Kx-ClRCLES ...... 100 ORTHOCENTRE ........ 101 CENTROID ..... . 103 v RIDERS ON BOOK IV PROPORTION ........ 105 SIMILAR TRIANGLES ....... Ill vii Vlll CONTENTS PACH8 RECTANGLE PROPERTIES OF A CIRCLE . . . .120 AREAS AND VOLUMES (SIMILAR FIOTJRKB, SOLIDS) . . ,126 THE BISECTOR OF AN ANGLE OF A TRIANGLE . . .131 CONSTRUCTION EXERCISES BOOK I USE OF INSTRUMENTS . . . . . .13.5 DRAWING TO SUALK ....... 145 MISCELLANEOUS I ....... 148 TRIANGLES, PARALLELOGRAMS, ETC . ... 150 MISCELLANEOUS II . . . . . . .153 CONSTRUCTION EXERCISES BOOK II AREAS ......... 155 SUBDIVISION OF A LINE .... .158 CONSTRUCTION EXERCISES BOOK 111 CIRCLES . . . . . . . .161 MISCELLANEOUS III . . . . . . .172 CONSTRUCTION EXERCISES BOOK IV PROPORTION, SIMILAR FIGURES ..... 175 MEAN PROPORTIONAL . . . . . . .178 MISCELLANEOUS IV ....... 180 REVISION PAPERS, I-L 181 PROOFS OF THEOREMS BOOK I ...... .205 BOOK II ........ 218 BOOK III ........ 234 BOOK IV ........ 254 PROOFS OF CONSTRUCTION BOOK I ........ 267 BOOK II 276 BOOK III ........ 280 BOOK IV ........ 289 NOTES 305 GLOSSARY AND INDEX 313 ANSWBKS 315 A CONCISE GEOMETRY RIDERS ON BOOK I ANGLES AT A POINT THEOREM 1 (i) If a straight line CE cuts a straight line ACB at C, then (ii) If lines CA, CB are drawn on opposite sides of a line CE such that L ACE + L. BCE= 180, then ACB is a straight line. C Fir,. 1. THEOREM 2 If two straight lines intersect, the vertically opposite angles are equal. 2 CONCISE GEOMETRY ANGLES AT A POINT EXERCISE I 1. What are the supplements of 20, 150, 27* 45', 92 10' 1 2. What are the complements of 75, 30 30', 10 48"? 3. A wheel has six spokes, what is the angle between two adjacent spokes ? 4. Guess the sizes of the following angles : FIG. 3. 5. What is the least number of times you must turn through 17 in order to turn through (i) an obtuse angle, (ii) a reflex angle, (iii) more than oue complete revolution 1 6. What is the angle between N.K. and S.E. 1 7. What is the angle between S.S.W. and E.N.E. ? 8. What is the angle between (i) 12 N. of W. and 5 E. of N. ; (ii) S.W. and E.S.E. ; (iii) 22 S. of W. and 9 N. of E. ? 9. Through what angle does the minute hand of a clock turn in 15 minutes, 5 minutes, 20 minutes, 50 minutes, 2 hours 45 minutes t 10. Through what angle does the hour hand of a clock turn in 40 minutes, 1 hour ^ 11. Through what angle has the hour hand of a clock turned, when the minute hand has turned through 30 1 12. What is the angle between the hands of a clock at (i) 4 o'clock, (ii) ten minutes past four 1 13. A wheel makes 20 revolutions a minute, through what angle does a spoke turn each second ? 14. What equation connects x and y if x* and y are (i) comple- mentary, (ii) supplementary ? ANGLES AT A POINT 3 15. The line OA cuts the line BOC at O; if LAOB = 2< AOC, calculate <_AOB. 16. What angle is equal to four times its complement? 17. A man watching a revolving searchlight notes that he is in the dark fou** times as long as in the light, what &ngle of country does the searchlight cover at any moment ? 18. The weight in a pendulum clock falls 4 feet in 8 days; through what angle does the hour hand turn when the weight falls 1 inch *? 19. What is the reflex angle between the directions S.W. and N.N.W. ? 20. If the earth makes one complete revolution every 24 hours, through what angle does it turn in 20 minutes ? 21. The longitude of Boston is 71 W., and of Bombay is 73 E., what is their difference of longitude ? 22. The latitude of Sydney is 33 S., and of New York is 41 K, what is their difference of latitude 1 23. Cape Town has latitude 33 40' S. and longitude 18 30' E., Cologne has latitude 50 55' N. and longitude 7 E., what is their difference of latitude and longitude ? 24. z.POQ = 2^, L.QOR = 3#, ^POR = 4#; find x. 25 OP, OQ, OR, OS are 4 lines in order such that L POQ = 68 Q , L QOR * 53, L ROS = 129 ; find L SOP. Find also the angle between the lines bisecting L POS, L QOR. 26. OA, OB, OO are 3 lines in order such that L AOB = 54, L BOC = 24 ; OP bisects L AOC ; find L POB. 27. CD is perpendicular to ACB ; CE is drawn so that L. DOE = 23 ; find the difference between L ACE and L BCE. What is their sum 1 CONCISE GEOMETRY 28. Given ZAOD = 145, ZBOC - 77, and ZAOB- ZCOD; calculate Z.AOC (Fig. 5). FIG. 5. 29. 30. OA, OB, OC, OD, OE, OF are 6 linen in order such that ZAOB -43, Z BOG -67, Z COD -70, Z DOE -59, Z EOF =^51; prove that AOD and COF are straight lines. Calculate the angle between the lines bisecting ZAOF and ZBOC. ZAOB = 38; AO is produced to C; OP bisects ZBOC; calculate reflex angle AOP. 31. OA, OB, OC, OD are 4 lines in order such that Z AOC = 90 = Z BOD ; if Z BOC = #*, calculate Z AOD. 32. Two lines AOB, COD intersect at O; OP bisects ZAOC; if ZBOC = #, calculate ZDOP. 33. OA, OC make acute angles with OB on opposite sides; OP bisects Z BOC ; prove Z AOB + Z AOC = 2 Z AOP. The line OA cuts the line BOC at O; OP bisects ZAOB; OQ bisects Z AOC ; prove Z POQ = 90. OA, OB, OC, OD are 4 lines in order such that ZAOB = ZCOD and Z BOC = Z AOD ; prove that AOC is a straight line. Given ZAOB= ZDOC, and that OP bisects ZAOD (see Fig. 6), if PO is produced to Q, prove that QO bisects Z BOC. 34. 35. 36. ANGLES AND PARALLEL LINES ANGLES AND PARALLKL LINES THEOREM 5 In Fig. 7, (i) If Z.PBC- Z.BCS, then PQ is parallel to RS. ii) If Z. ABQ = /. BCS, then PQ is parallel to RS. ii) If Z.QBC+ Z. BCS = 180, then PQ is parallel to RS. -Q 'D FIG. 7. THEOREM 6 In Fig. 7, If PQ is parallel to RS, Then (i)/.PBC- Z.BCS (alternate angles). (ii)/.ABQ~ /.BCS (corresponding angles), (iii) Z. QBC + /. BCS = 180. 6 CONCISE GEOMETRY ANGLES AND PARALLEL LINES EXERCISE II 1. In the following figures, a line cuts two parallel lines. What equations connect the marked angles 1 Give reasons. FIG. 8. 2. The following figures contain pairs of parallel lines. What equations connect the marked angles 1 Give reasons. J iL. 3. (i) If one angle of a parallelogram is 60, find its other (ii) If one angle of a parallelogram is 90, find its other angles. 4. If AB is parallel to ED, sec Fig. 10, prove that ^.BCD = D E FIG. 10. 5. The side AB of the triangle ABC is produced to D; BX is drawn parallel to AC; z.BAC = 32, z.BCA = 47; find the remaining angles in the figure and the value of L BAC -f _ABC. ANGLES AND PARALLEL LINES 7 6. If AB is 'parallel to DE, see Fig. 11, prove that Z.ABC + /.ODE =180+ Z-BCD. Dr E FIG. 11. [Draw CF parallel to DE.] 7. In Fig. 12, prove that AB is parallel to ED. D E FIG. 12. 8. In Fig. 13, if /.ABC = 74, /.EDO = 38, L BCD = 36, prove ED is parallel to AB. A/ FIG. 13. 9. ABCD is a quadrilateral ; if ABjis parallel to DC, prove that L DAB - L DCB = L ABC - L. ADC. 10. In Fig. 14, if AB is parallel to DE, prove that # + y - z equals two right angles. B _ A * ^"*r-^ FIG. 14. 8 CONCISE GEOMETRY 11. A line AC cuts two parallel lines AB, CD; B and D are on the same side of AC ; the lines bisecting the angles CAB, ACD meet at O ; prove that /. AOC == 90. 12. If two straight lines are each parallel to the same straight line, prove that they are parallel to each other. ANGLES OF A TRIANGLE ANGLES OF A TRIANGLE VNU OTHER RECTILINEAL FIGURES THEOKEM 7 (i) If the side BC of the triangle ABC is produced to D, L ACD = L BAG + L ABC. (ii) In the AABC, Z.ABC+ Z.BCA+ L CAB = 180. C FIG. 15. THEOKEM 8 (i) The sum of the interior angles of any convex polygon which has n sides is 2n 4 right angles. (ii) If the sides of a convex polygon are produced in order, the sum of the exterior angles is 4 right angles. 10 CONCISE GEOMETRY ANGLES OF A TRIANGLE AND OTHER RECTILINEAL FIGURES EXERCISE III 1. In a right-angled triangle, one angle is 37, what is the third angle ? 3. Two angles of a triangle are each 53, what is the third angle 1 . If L BAC = 43 and L ABC = 109, what is L ACB 1 A. The side BC of the triangle ABC is produced to D ; L. ABD = 19, L ACD 37, what is L BAC ? 5. In the quad. ABCD, /.ABC =112, /.BCD = 75, /.DAB = 51, what is ^.CDA? jB. ABCD is a straight line and P a point outside it ; L PBA = 110, /.PCD =163, find L BPC. 7. Three of the angles of a quad, are equal ; the fourth angle is 120; find the others. 8. Can a triangle be drawn having its angles equal to (i) 43, 64, 15. What is the remaining angle of a triangle, if two of its angles are (i) 120, 40; (ii) 50, x '; (iii) 2x, 3^; (iv) x +10, 20 - x degrees 1 ). The angles of a triangle are #, 2#, 2# ; find x. ^H. If in the triangle ABC, L BAC L BCA + L ABC, find /.BAC. /I2. If A, B, C are the angles of a triangle and if A - B = 15, . B~C = 30, find A. 13. The angles of a five-sided figure are #, 2#, a? + 30, x -10, x + 40 degrees, find x. /1 4. The angles of a pentagon are in the ratio 1:2:3:4:5; find them. 5. In AABC, /.ABC = 38, /.ACB = 54; AD is perpendicular to BC ; AE bisects /. BAC, find L EAD. 43. r In AABC, /.BAC = 74, ABC = 28; BC is produced to X; the lines bisecting /.ABC and LACX meet at K. Find /.BKC. ANGLES OF A TEIANGLE 11 vt7. In AABC, i ABC = 32, L BAC = 40; find the angle at which the bisector of the greatest angle of the triangle cuts the opposite side. 18. In AABC, L ABC =110, <LACB = 50; AD is the per- pendicular from A to CD produced \ prove that L DAB = 19. The base BC of AABC is produced to D ; if L ABC= L ACB and if L ACD = o?, calculate L BAC. 20. In the quad. ABCD, L ABC -140, L ADC = 20; the lines bisecting the angles BAD, BCD meet at O; calculate 21. In AABC, the bisector of L BAC cuts BC at D , if L ABC = a? and L. ACB = y, calculate L ADC. >22wlf the angles of a quad, taken in order are in the ratio 1:3:5:7, prove that two of its sides are parallel. -3. Each angle of a polygon is 140; how many sides has it? -24. Find the sum of the interior angles of a 12-sided convex polygon. 25. Find the interior angle of a regular 20-sided figure. 26. Prove that the sum of the interior angles of an 8-sided convex polygon is twice the sum of those of a pentagon. 27. Each angle of a regular polygon of x sides is f of each angle of a regular polygon of y sides ; express y in terms of #, and find any values of #, y which will fit. 28. The sum of the interior angles of an rc-sided convex polygon is double the sum of the exterior angles. Find n. 29. In Fig. 16, prove that x = a 4- b - y. FIG. 16. 12 CONCISE GEOMETRY 30. In Fig. 17, prove that x-y^a-b. FIG. 17. 31. In Fig. 18, express x in terms of a, 6, c. FIG. 18. 32. In Fig. 19, express x in terms of a, ft, c. FIG. 19. 33. In Fig, 20, express x in terms of a, 6, c. FIG. 20. ANGLES OF A TRIANGLE 13 34. If, in Fig. 21, x -fy = 3z, prove that the triangle is right- angled. 35. Prove that the reflex angles in Fig. 22 are connected by the relation a + 6 = x + y. 36. D is a point on the base BC of the triangle ABC such that L DAG = L ABC, prove that L ADC = L BAC. $7. The diagonals of the parallelogram ABCD meet at O, prove that L AOB = L ADB + L ACB. IBS. If, in the quadrilateral ABCD, AC bisects the angle DAB and the angle DCB, prove that L. ADC = L ABC. $9. ABC is a triangle, right-angled at A ; AD is drawn perpen- dicular to BC, prove that L DAC = L ABC. 40. ABCD is a parallelogram, prove that the lines bisecting the angles DAB, DCB are parallel. "il. In the AABC, BE and CF are perpendiculars from B, C to AC, AB ; BE cuts CF at H ; prove that L CHE - L BAC. ^2. If, in the quadrilateral ABCD, ^.ABC= _ ADC and L BCD = L BAD, prove that ABCD is a parallelogram. 43. If in the AABC the bisectors of the angles ABC, ACB meet at I, prove that L BIC = 90 + \ L BAC. 14 CONCISE GEOMETRY The side BC of the triangle ABC is produced to D ; CP is drawn bisecting L ACD ; if L CAB = L CBA, prove that CP is parallel to AB. 45. The side BC of A ABC is produced to D ; the lines bisecting L ABC, L ACD meet at Q ; prove that L BQC = \ L BAG. The base BC of A ABC is produced to D; the bisector of L BAG cuts BC at K ; prove L ABD -f L ACD = 2 L AKD. The sides AB, AC of the triangle ABC are produced to H, K; the lines bisecting L HBC, L KGB meet at P; prove that ^48. P is any point inside the triangle ABC, prove that LBPO/.BAC. y49. In the quadrilateral ABCD, the lines bisecting L ABC, L BCD meet at P, prove that L. BAD + L CDA = 2 L. BPC. CONGRUENT TRIANGLES Given a triangle ABC, what set of measurements must be made in order to copy it '? 1. Measure AB, AC, /.BAG. This is enough to fix. the size and shape of the triangle. Therefore all triangles drawn to these measurements will be congruent to A ABC and to each other. This result is given as Theorem 3. 2. Measure BC, Z.ABC, Z.ACB. This also fixes the triangle. [Theorem 9.] 3. Measure BC, CA, AB. This also fixes the triangle. [Theorem 11.] ISOSCELES TRIANGLES 15 ISOSCELES TRIANGLES AND CONGRUENT TRIANGLES (FIRST SECTION) THEOHEM 3 In the triangles ABC, PQR, If AB-PQ, AC = PR, /.BAG- -. QPR, Then AABC==APQR. B C Q FIG. 23. THEOREM 9 In the triangles ABC, PQR, (i) If BC = QR, /-ABC= L PQR, A.ACB = L PRQ, Then AABC = APQR. (ii) If BC = QR, L ABC = L PQR, L BAG = u QPR, Then AABC = APQR. THEOREM 10 ABC is a triangle. (i) If AB = AC, then L ACB = L ABC (ii) If L ACB - L ABC, then AB AC. B FIG. 24. 16 CONCISE GEOMETRY ISOSCELES TRIANGLES AND CONGRUENT TRIANGLES (FIBST SECTION) EXERCISE IV \. The vortical angle of an isosceles triangle is 110; what are the base angles 1 2. One base angle of an isosceles triangle is 62 ; what is the vertical angle *! 3. Find the angles of an isosceles triangle if (i) the vertical angle is double a base angle, (ii) a base angle is double the vertical angle. 4. In the triangle ABC, L BAG = 2 L ABC and L. ACB - L ABC 36 ; prove that the triangle is isosceles. 5. A, B, C are three points on a circle, centre O ; L AOB = 100, L BOC = 140, calculate the angles of the triangle ABC. 6. In Fig. 25, if AB = AC, find x in terms of y. 7. D is a point on the base BC of the isosceles triangle ABC such that BD = BA; if Z-BAD = # and <LDAC=y, express x in terms of y. 8. ABCDE is a regular pentagon, prove that the line bisecting the angle BAC is perpendicular to AE. 9. In the triangle ABC, AB = AC ; D is a point in AC such that AD = BD = BC. Calculate Z.BAC. 10. If the line PQ bisects AB at right angles, prove that PA = PB. '11. Two unequal lines AC, BD bisect each other, prove that AB = CD. 12. In the quadrilateral ABCD, AB is equal and parallel to DC; prove that AD is equal and parallel to BC. /i3. A line AP is drawn bisecting the angle BAC; PX, PY are the perpendiculars from P to AB, AC ; prove that PX = PY. ISOSCELES TRIANGLES 17 14. D is the mid-point of the base BC of the triangle ABC, prove that B and C are equidistant from the line AD. Yb. A straight line cuts two parallel lines at A, B ; C is the mid-point of AB ; any line is drawn through C cutting the parallel lines at P, Q ; prove that PC = CQ. 16. If the diagonal AC of the quadrilateral ABCD bisects the angles DAB, DCB, prove that AC bisects BD at right angles. 17. ABCD is a quadrilateral; E, F arc the mid-points of AB, CD; if L AEF = 90 = L EFD, prove that AD = BC. 18. The diagonals of a quadrilateral bisect each other at right angles, prove that all its sides are equal. 19. Two lines POQ, ROS bisect each other, prove that the triangles PRS, PQS are equal in area. 20. Two lines POQ, ROS intersect at O ; SP and QR are produced to meet at T ; if OP = OR and OS = OQ, prove TS - TQ. 21. ABC is an equilateral triangle; BC is produced to D so that BC = CD ; prove that /. BAD = 90. -82. In the AABC, AB = AC; AB is produced to D so that BD = BC ; prove that L ACD = 3 ADC. 33. P is a point on the line bisecting /. BAG ; through P, a line is drawn parallel to AC and cutting AB at Q; prove AQ = QP. #4. In AABC, AB = AC ; D is a point on AC produced such that BD = BA; if ZCBD = 36, prove BC = CD. 5. If in Fig. 26, AB = AC and CP - CQ, prove /. SRP - 3 Z. RPC. FIG. 26. o. The base BC of the isosceles triangle ABC is produced to D ; the lines bisecting Z.ABC and ^lACB meet at I; prove Z.ACD=Z.BIC. 2 18 CONCISE GEOMETRY '27. In the quadrilateral ABCD, AB-AD and Z ABC- Z.ADC, prove CB = CD. } 28. ABC is an acute-angled triangle ; AB< AC ; the circle, centre A, radius AB cuts BC at D, prove that /.ABC + /.ADC = 180. ^9. A, B, C are three points on a circle, centre O ; prove Z. ABC = ZLOAB+/.OCB. <#0. AB, AC are two chorda of a circle, centre O ; if /. BAC - 90, prove that BOC is a straight line. 31. In the A ABC, AB = AC; the bisectors of the angles ABC and ACB meet at I, prove that IB ~ 1C. 32. AD is an altitude of the equilateral triangle ABC ; ADX is another equilateral triangle, prove that DX is perpendicular to AB or AC. '33r~^C is the base of an isosceles triangle ABC ; P, Q are point* on AB, AC such that AP = PQ = QB = BC ; calculate L. BAC. 34. D is the mid-point of the base BC of the triangle ABC ; if AD = DB, prove L BAC = 90. 35. In the quadrilateral ABCD, AB = CD and ^ABC= L DCB, prove L. BAD -= L CDA. '36. In the A ABC, AB>AC; D is a point on AB such that AD = AC ; prove L ABC + L ACB 2 L ADC. "37. The triangle ABC is right-angled at A; AD is the per- I>endicular from A to BC; P is a point on CB such that CP = CA ; prove AP bisects L BAD. 38. The vertical angles of two isosceles triangles are supplement- ary ; prove that their base angles are complementary. 39. Draw two triangles ABC, XYZ which are such that AB XY, AC XZ, L ABC = L XYZ but are not congruent. Prove '40. In the A ABC, AB = AC; P is any point on BC produced; PX, PY are the perpendiculars from P to AB, AC produced ; prove L. XPB = L. YPB. 41. ABC is any triangle; ABX, ACY are equilateral triangles external to ABC ; prove CX = BY. 42. OA = OB = OC and L BAC is acute ; prove L BOC == 2 L BAC. 43. In the A ABC, AB = AC ; AB is produced to D ; prove L. ACD - L ADC 2 L BCD. ISOSCELES TRIANGLES 1!) 44. D is a point on the side AB of AABC such that AD = DC = CB ; AC is produced to E ; prove L ECB = 3 L ACD. 45. In the A ABC, L BAG is obtuse ; the perpendicular bisectors of AB, AC cut BC at X, Y ; prove L XAY - 2 - BAG - 180. 46. In the A ABC, AB = AC and/- BAC>60; tho perpendicular bisector of AC meets BC at P ; prove L. APB = 2 L. ABP. 47. D is the mid-point of the -side AB of A ABC ; the bisector of L ABC cuts the line through D parallel to BC at K ; prove 48. In the A ABC, L_ BAG = 90 and AB=AC; P, Q are points on AB, AC such that AP = AQ ; prove that the perpendicular from A to BQ bisects CP. 49. X, Y are the mid-points of the sides AB, AC of the A ABC ; P is any point on a line through A parallel to BC ; PX, PY are produced to meet BC at Q, R ; prove QR = BC. 50. ABC is a triangle; the perpendicular bisectors of AB, AC meet at O ; prove OB = OC. 51. ABC is a triangle; the lines bisecting the angles ABC, ACB meet at I ; prove that the perpendiculars from I to AB, AC are equal. 52. The sides AB, AC of the triangle ABC are produced to H, K ; the lines bisecting the angles HBC, KCB meet at I ; prove that the perpendiculars from I to AH, A K are equal. 53. Two circles have the same centre ; a straight line PQRS cuts one circle at P, S and the other at Q, R ; prove PQ = RS. 54. ABC is a A ; a line AP is drawn on the same side of AC as B, meeting BC at P, such that /. CAP = _ ABC ; a line AQ is drawn on the same side of AB as C, meeting BC at Q, such that L BAQ = L ACB j prove AP = AQ. 55. The line joining the mid-points E, F of AB, AC is produced to Q so that EF - FG ; prove that BE is equal and parallel toCG. 56. In the 5-sided figure ABODE, the angles at A, B, C, D are each 120 ; prove that AB + BC = DE. 57. ABC is a triangle ; lines are drawn through C parallel to the bisectors of the angles GAB, CBA to meet AB produced in D, E ; prove that DE equals the perimeter of the triangle ABC. 20 CONCISE GEOMETRY 58. AB, BC, CD are chords of a circle, centre O ; if Z AOB = 108 9 L BOG = 60, L COD = 36, prove AB = BC + CD. [From BA cut off BQ equal to BO : join OQ.] 59. In the triangles ABC, XYZ, if BC = YZ, ZABC=/.XYZ, AB + AC = XY, prove L BAG = 2 /. YXZ. 60. In the A ABC, AB = AC and Z.ABC = 2Z.BAC; BC is pro- duced to D so that /. CAD = \ /. BAG ; CF is the perpendic- ular from C to AB ; prove AD = 2CF. CONGRUENT TRIANGLES 21 CONGRUENT TRIANGLES (SECOND SECTION), PARALLELOGRAMS, SQUARES, ETC. THEOREM 11 In the triangles ABC, XYZ, If AB = XY, BC = YZ, CA = ZX, Then C Y FIG. 27. THEOREM 12 In the triangles ABC, XYZ, If /. ABC = 90 = Z.XYZ, AC = XZ, AB = XY, Then AABC = AXYZ. THEOREM 13 If A BCD is a parallelogram, Then (i) AB = CD and AD = BC. (ii) L DAB = L DCB and L ABC == L ADC. (iii) BD bisects ABCD. B FIG. 28. 22 CONCISE GEOMETRY THEOREM 14 If the diagonals of the parallelogram ABCD intersect at O, Then AO = OC and BO = OD. THEOREM 16 If AB is equal and parallel to CD, Then AC is equal and parallel to BD. C D FIG. 30. DEFINITIONS. A parallelogram is a four-sided figure whose opposite sides are parallel. A rectangle is a parallelogram, one angle of which is a right angle. A square is a rectangle, having two adjacent sides equal. A rhombus is a parallelogram, having two adjacent sides equal, but none of its angles right angles. A trapezium is a four-sided figure with one pair of opposite sides parallel. 'CONGRUENT TRIANGLES 23 CONGRUENT TRIANGLES (SECOND 8wr;*>jr), PARALLELOGRAMS, SQUARES, Era EXERCISE V 1. Prove that all the sides of a rhombus are equal. 2. Prove that the diagonals of a rectangle are equal. 3. Prove that the diagonals of a rhombus intersect at right angles. 4. Prove that the diagonals of a square are equal and cut at right angles. 5. The diagonals of the rectangle ABCD meet at O ; Z. BOC = 44 ; calculate Z.OAD. 6. Prove that a quadrilateral, whose opposite sides are equal, is a parallelogram. 7. ABCD is a rhombus ; /.ABC = 56; calculate Z. ACD. 8. ABCD is a parallelogram ; prove that B and D are equidistant from AC. 9. X is the mid-point of a chord AB of a circle, centre O ; prove /. OX A = 90. 10. The diagonals of the parallelogram ABCD cut at O ; any line through O cuts AB, CD at X, Y ; prove XO = OY. 11. Two straight lines POQ, ROS cut at O ; if PQ = RS and PR = QS, prove /. RPO = /. QSO. 12. In the quadrilateral ABCD, AB = CD and AC = BD; prove that AD is parallel to BC. 13. E is a point inside the square ABCD; a square AEFG is described on the same side of AE as D ; prove BE DG. 14 ABC is any triangle; BY, CZ are lines parallel to AC, AB cutting a line through A parallel to BC in Y, Z; prove AY = AZ. 15. ABCD is a parallelogram ; P is the mid-point of BC ; DP and AB are produced to meet at Q ; prove AQ = 2AB. 16. ABCD, ABXY are two parallelograms; BC and BX are I different lines ; prove that DCXY is a parallelogram. 17. Two unequal circles, centres A, B, intersect at X, Y; prove that AB bisects XY at right angles. 18. The diagonals of a square ABCD cut at O; from AB a part AK is cut off equal to AC ; prove L AOK = 3 /. BOK. 24 CONCISE GEOMETRY 19. ABCD is a straight line such that AB = BC = CD; BCPQ is a rhombus ; prove that AQ is perpendicular to DP. 20. ABCD is a parallelogram ; the bisector of Z. ABC cuts AD at X ; the bisector of L BAD cuts BC at Y ; prove XY = CD. 21. ABCD is a parallelogram such that the bisectors of Z. s DAB, ABC meet on CD ; prove AB = 2BC. 22. In AABC, ZBAC^90; BADH, ACKE are squares outside the triangle ; prove that HAK is a straight line. 23. The diagonals of the rectangle ABCD cut at O; AO>AB; the circle, centre A, radius AC cuts AB produced at E ; if L AOB = 4 Z.BOE, calculate ,/BAC. 24. ABC is an equilateral triangle ; a line parallel to AC cuts BA, BC at P, Q; AC is produced to R so that BQ = CR; prove that PR bisects CQ. 25. P is one point of intersection of two circles, centres A, B ; AQ, BR are radii parallel to and in the same sense as BP, AP ; prove that QPR is a straight line. 26. In AABC, L BAG = 90; ABPQ, ACRS, BCXY are squares outside ABC ; prove that (i) BQ is parallel to CS; (ii) BR is perpendicular to AX. 27. ABC is a triangle; the bisectors of Z.s ABC, ACB meet at I ; prove IA bisects /.BAG. [From I drop perpendiculars to AB, BC, CA.] 28. In AABC, Z.BAC = 90; BCPQ, ACHK are squares outside ABC ; AC cuts PH at D ; prove AB = 2CD and PD = DH. 29. In AABC, AB = AC; P is any point on BC; PX, PY are the perpendiculars from P to AB, AC; CD is the per- pendicular from C to AB ; prove PX + PY = CD. 30. In AABC, AB = AC; P is a variable point on BC; PQ, PR are lines parallel to AB, AC cutting AC, AB at Q, R ; prove that PQ -f PR is constant. 31. H, K are the mid-points of the sides AB, AC of AABC; HK is joined and produced to X so that HK = KX; prove that (i) CX is equal and parallel to BH ; (ii) HK= JBC and HK is parallel to^BC. RIDERS ON BOOK II AREAS THEOREM 16 (i) If ABCD and ABPQ are parallelograms on the same base and between the same parallels, their areas are equal, (ii) If BH is the height of the parallelogram ABCD, area of ABCD = AB . BH. FIG. 32. THEOREM 17 If AD is an altitude of the triangle ABC, areaof ABC = iAD.BC. D C FIG. 38. 2,. 26 CONCISE GEOMETRY THEOREM 18 (i) If ABC and ABD arc triangles on the same base and between the same parallels, their areas are equal, (ii) If the triangle ABC, ABD are of equal area and lie on the same side of the common base AB, they are between the same parallel**, i.e. CD is parallel to AB. THEOREM 19(1) If the triangle ABC and the parallelogram ABXY are on the same base AB and between the same parallels, area of ABC = area of ABXY. AREAS 27 THEOREM 19(2) (i) Triangles (or parallelograms) on equal bases and between the same parallels are equal in area, (ii) Triangles (or parallelogram i) of equal area, which are on equal bases in the same straight line and on the same side of it, are between the same parallels. D C W Z B X FIG. 36. MENSURATION THEOREMS (i) If the lengths of the parallel sides of a trapezium are a inches and b inches, and if their distance apart is h inches, area of trapezium = \h (a -f- b) sq. inches. b FIG. 37. (ii) If the lengths of the sides of a triangle are a, b, c inches and if $== (a + b + c), area of triangle = /s (a -)(*- b) (* - c) sq. inches. 28 CONCISE GEOMETRY AREAS TRIANGLES, PAKALLELOGKAMS, ETC. EXERCISE VI In Fig. 38, AD, BE, CF are altitudes of the triangle ABC. A 1. In A ABC, L ABC = 90, AB - 3", BC = 5* ; find area of ABC. 2. In Fig. 38, AD - 7", BC = 5" ; find area of ABC. 3. In Fig. 38, BE = 5", CF = 6*, AB = 4"; find AC. 4. In Fig. 38, AD = 6#*, BE = 4#", CF = 3#*, and the perimeter of ABC is 18*. Find BC. 5. In quad. ABCD, AB = 12*, BC = 1", CD = 9", DA = 8*, L ABC = L ADC = 90 ; find the area of ABCD. 6. In quad. ABCD, AC = 8*, 60 = 11", and AC is perpendicular to BD ; find the area of ABCD. 7. Find the area of a triangle whose sides are 3", 4", 5". In Fig. 39, ABCD is a parallelogram ; AP, AQ are the per- pendiculars to BC, CD. A B FIG. 39. 8. In Fig. 39, AB = 7*, AQ = 3* ; find the area of ABCD. 9. In Fig. 39, AB = 5* AD = 4", AP = 6* ; find AQ. 10. In Fig. 39, AP = 3*, AQ = 2 /; , and perimeter of ABCD is find its area. AREAS 29 11. In quad. ABCD, BC = 8", AD = 3", and BC is parallel to AD; if the area of AABC is 18 sq. in., find the area of AABD. 1 2. In quad. ABCD, AB = 5", BC = 3", CD = 2", L ABC = L BCD = 90 ; find the area of ABCD. . 13. In Fig. 38, AB = 8", AC = 6", BE -5"; find CF. 14. The area of AABC is 36 sq. cms., AB = 8 cms., AC = 9 cms., D is the mid-point of BC; find tho lengths of the per- pendiculars from D to AB, AC. 15. In the parallelogram ABCD, AB = 8", BC = 5"; the per- pendicular from A to CD is 3" ; find the perpendicular from B to AD. 16. Find the area of a rhombus whose diagonals are 5", 6*. 17. In AABC, L ABC = 90, AB = 6", BC = 8", CA = 10" ; D is the mid-point of AC. Calculate the lengths of the per- pendiculars from B to AC and from A to BD. 18. On an Ordnance Map, scale 6 inches to the mile, a football field is approximately a square measuring \ inch each way. Find the area of the field in acres, correct to ^ acre. 19. Fig. 40 represents on a scale of 1" to the foot a trough and the depth of water in it. The water is running at 4 miles an hour ; find the number of gallons which pass any point in a minute, to nearest gallon, taking 1 cub. ft. = 6 J gallons. FIG. 40. 20. Fig. 41 represents on a scale of 1 cm. to 100 yds. the plan of a field ; find its area in acres correct to nearest acre. FIG. 41. 30 CONCISE GEOMETRY 21. Fig. 42 represents the plan and elevation of a box on a scale of 1 cm. to 1 ft. (i) Find the volume of the box. (ii) Find the total area of its surface. FIG. 42. 22. The diagram (Fig. 43), not drawn to scale, represents the plan ol an estate of^6| acres. The measurements given are in inches. On what scale (inches to the mile) is it drawn 1 The dotted line PQ divides the estate in half ; find AQ. 4 A ! 4 FIG. 43. 23. Find the area of ABCD (Fig. 44) in tormstof #, y, p, </, r. AREAS 31 24. ABC is inscribed in a rectangle (Fig. 45) ; find the area of ABC in terms of p, r/, r, s. Fir.. 45. 25. In Fig. 46 /.ABC- /. BCD = 90 J . Find the length of the perpendicular from C to AD in terms of p, q, r. B FIG. 46. 26. In Fig. 47 OB is a square, side 4*; OA=12", 00 = 6*. Calculate areas of AOAB, AOBC, AAOC, and prove that ABC is a straight line. C B FIG. 47. 27. In AAOB, OA=-a, OB = b, Z.AOB = 90; P is a point on AB j PH, PK are the perpendiculars from P to OA, OB ; ; prove - + a o 28. P, Q are points on the sides AB, AD of the rectangle ABCD ; AB = ^, AD = y, PB=e, QD=/. Calculate area of PCQ in terms of e, /, #, y. 29. The area of a rhombus is 25 sq. cms., and one diagonal is half the other; calculate the length of each diagonal 32 CONCISE GEOMETRY 30. Find the area of the triangles whose vertices are : (i) (2,1); (2, 5); (4, 7). (ii) (3, 2); (5, 4); (4, 8). (iii)(l, 1);(5,2);<6,5). (iv) (0,0); (a,o); (b, c). (v)(0,0);(a,6);(c,d). 31. Find the area of the quadrilaterals whose vertices are : (i) (0,0); (3,2); (1,5); (0,7). (ii) (1,3); (3, 2); (5, 5); (2, 7). 32. Find in acres the areas of the fields of which the following field-book measurements have been taken : (1) to C 80 to B 50 YARDS to D 250 200 150 100 From A 40 to E (2) to C 60 to B 100 YARDS to D 300 220 50 to E 200 100 50 i 80 to F From A ! 33. Find from the formula [page 27] the area of the triangles whose sides are (i) 5 cms., 6 cms., 7 cms. (ii) 8*, 15*, 19*. Find also in each case the greatest altitude. 34. The sides of a triangle are 7", 8*, 10*. Calculate its shortest altitude. 35. AX, BY are altitudes of the triangle ABC; if AC = 2BC, prove AX = 2BY. 36. ABC is a A ; a line parallel to BC cuts AB, AC at P, Q ; prove AAPC = AAQB. 37. Two lines AOB, COD intersect at O; if AC is parallel to BD, prove AAOD = ABOC. 38. The diagonals AC, BD of ABCD are at right angles, prove that area of ABCD = \ AC . BD. 39. The diagonals of the quad. ABCD cut at O; if AAOB = AAOD, prove ADOC = ABOC. 40. In the triangles ABC, XYZ, AB = XY, BC = YZ, Z.ABC + Z.XYZ = 180, prove AABC = AXYZ. AREAS 33 41. P is any point on tne median AD of AABC; prove AAPB- AAPC. 42. ABCD id a quadrilateral ; lines are drawn through A, C parallel to BD, and through B, D parallel to AC ; prove that the area of the parallelogram so obtained equals twice the aroa of ABCD. 43. ABCD is a parallelogram ; P is any point on AD ; prove that APAB -f- APCD = APBC. 44. ABC is a straight line; O is a point outside it; prove AOAB __ AB AOBC~BC" 45. ABCD is a parallelogram ; P is any point on BC ; DQ is the perpendicular from D to AP; prove that the area of ABCD-DQ.AP. 46. ABCD is a parallelogram; P is any point on BD; prove APAB = APBC. 47. ABCD is a parallelogram; a line parallel to BD cuts BC, DC at P, Q ; prove AABP= AADQ. 48. AOB is an angle ; X is the mid-point of OB ; Y is the mid- point of AX ; prove AAOY = ABXY. 49. If in Fig. 48, AC is perpendicular to BD, prove area of ABCD - J AC . BD. FIG. 48. 50. ABCD is a quadrilateral; a line through D parallel to AC meets BC produced at P ; prove that AABP = quad. ABCD. 51. ABCD is a quadrilateral; E, F are the mid-points of A.B, CD; prove that AADE + ACBF ABCE -f AADF. 52. The diagonals of a quadrilateral divide it into four triangles of equal area ; prove that it is a parallelogram. 53. ABCD and PQ are parallel lines; AB = BCCD = PQ; PC cuts BQ at O ; prove quad. ADQP = 8 AOBC. 3 34 CONCISE GEOMETRY 54. X, Y are the mid-points of the sides AB, AC of AABC ; prove that AXBY = AXCY and deduce that XY is parallel to BC. 55. Two parallelograms ABCD, AXYZ of equal area have a common angle at A ; X lies on AB ; prove DX, YC are parallel. 56. The sides AB, BC of the parallelogram ABCD are produced to any points P, Q ; prove A PCD = AQAD. 57. ABC is a A ; D, E are the mid-points of BC, CA ; Q is any point in AE; the line through A parallel to QD cuts BD at P ; prove PQ bisects AABC. 58. The medians BE, CF of AABC intersect at G; prove that ABGC = ABGA - AAGC. 59. In Fig. 49, the sides of AABC are equal and parallel to the sides of AXYZ ; prove that BAXY + ACZX = BCZY. FIG. 49. 60. ABP, AQB are equivalent triangles on opposite sides of AB ; prove AB bisects PQ. 61. ABCD is a parallelogram; any line through A cuts DC at Y and BC produced at Z; prove ABCY = ADYZ. 62. In Fig. 50, PR is equal and parallel to AB; PQAT and CQRS are parallelograms; prove they are equivalent. A R AREAS 35 63. BE, CF are medians of the triangle ABC and cut at G ; prove ABGC = quad. AEGF. 64. ABC, ABD are triangles on the same base and between the same parallels ; BC cuts AD at O ; a line through O parallel to AB cuts AC, BD at X, Y ; prove XO = OY. i i i 65. In Fig. 51, APQR is a square; prove B K -. AB AC 66. ABCD is a quadrilateral ; AB is parallel to CD ; P is the mid- point of BC ; prove ABCD = 2 AAPD. 67. ABCD is a parallelogram ; DC is produced to P ; AP cuts BD at Q ; prove A DQP - AAQB = ABCP. 68. In Fig. 52, ABCD is divided into four parallelograms ; prove POSD = ROQB. D 8 C ^~ 69. In Fig. 52, prove A APR + AASQ = AABD. 70. ABC is a A ; any three parallel lines AX, BY, CZ meet BC, CA, AB produced where necessary at X, Y v Z ; prove AAYZ 71. In ex. 70, prove AXYZ-2AABC. 72. ABCD is a parallelogram; AB is produced to E; P is any point within the angle CBE; prove APBD. 36 CONCISE GEOMETRY 73*. ABC is a A ; ACPQ, BCRS are parallelograms outside ABC ; QP, SR are produced to meet at O ; ABXY is a parallelogram such that BX is equal and parallel to OC ; prove that ACPQ 4- BCRS = ABXY. A P B 74*. In Fig. 53, ABCD is divided into four parallelograms, prove that SOQD - BPOR = 2 AAOC, 75*. P is a variable point inside a fixed equilateral triangle ABC ; PX, PY, PZ are the perpendiculars from P to BC, CA, AB ; prove that PX -f PY -f PZ is constant. 76*. In A ABC, L ABC = 90; DBC Is an equilateral triangle out- side ABC ; prove AADC - ADBC = \ AABC. 77*. In AABC, Z. BAC - 90 \ X, Y, Z are points on AB, BC, CA such that AX YZ is a rectangle and AX |AB ; prove AX YZ = fAABC. 78*. Two fixed lines BA, DC meet when produced at O ; E, F are points on OB, CD such that OE = AB, OF = CD; P is a variable point in the angle BOD such that APAB-f APCD is constant ; prove that the locus of P is a line parallel to EF. 79*. G, H are the mid-points of the diagonals AC, BD of the quadri- lateral ABCD; AB and DC are produced to meet at P; prove quad. PYTHAGORAS' THEOREM 37 PYTHAGORAS' THEOREM THEOREM 20 If, in the triangle ABC, Z. BAG - 90", Then BA 2 + AC 2 = BC-. A C Fio. 54. THEOKEM 21 If, in the triangle ABC, BA 2 + AC 2 = BC 2 , Then ^.BAC-90 . 38 CONCISE GEOMETRY PYTHAGOKAS' THEOREM EXERCISE VII 1. In Fig. 54, AB = 5", AC =12", calculate BC. 2. In Fig. 54, AC = 6", BC= 10", calculate AB. 3. In Fig. 54, AB = 7", BC = 9", calculate AC. 4. In A ABC, AB = AC = 9", 60 = 8", calculate area of A ABC. 5. In AABC, AB = AC-13", BC = 10", calculate the length of the altitude BE. 6. Find the side of a rhombus whose diagonals are 6 3 10 cms. 7. A kite at P, flown by a boy at Q, is vertically above a point R on the same level as Q ; if PQ = 505', QR = 456', find the height of the kite. 8. In AABC, AC = 3", AB = 8", L ACB = 90 ; find the length of the median AD. 9. AD is an altitude of AABC ; AD = 2", BD = Y y DC = 4"; prove 10. ABCD is a parallelogram; AC = 13", BD-S calculate area of ABCD. 11. A gun, whose effective range is 9000 yards, is 5000 yards from a straight railway ; what length of the railway is com- manded by the gun ? 12. The lower end of a 20-foot ladder is 10 feet from a wall; how high up the wall does the ladder reach \ How much closer must it be put to reach one foot higher ? 13. An aeroplane heads due North at 120 miles an hour in an east wind blowing at 40 miles an hour ; find the distance travelled in ten minutes. 14. A ship is steaming at 15 knots and heading N.W. ; there is a 6-knot current setting N.E. ; how far will she travel in one hour? 15. AB, AC are two roads meeting at right angles ; AB = 110 yards, AC = 200 yards ; P starts from B and walks towards A at 3 miles an hour ; at the same moment Q starts from C and walks towards A at 4 miles an hour. How far has P walked before he is within 130 yards of Q ? 16. Find the distance between the points (1, 2), (5, 4). PYTHAGORAS' THEOREM 39 17. Prove that the points (5, 11), (6, 10), (7, 7) lie on a circle whose centre is (2, 7) ; and find its radius. 18. The parallel sides of an isosceles trapezium are 5*, II", and its area is 32 sq. inches ; find the lengths of the other sides. 1 9. In AABC, L ABC = 90, L ACB = 60, AC = 8* ; find AB. 20. In AABC, L ABC = 90, L ACB = 60, AB = 5"; find BC. 21. In Fig. 55, AB = 2", BC = 4", CD = V ; iind AD. 90 B 1 ocT ID Flo. 55. 22. In quadrilateral ABCD, AB = 5", BC = 12", 00 = 7"; Z.ABC = L BCD = 90 ; P, Q are points on BC such that L APD = 90= Z.AQD ; calculate BP, BQ. 23. In Fig. 56, AC = CB = 12", CD = 8*, ZACD = 90; find radius of circular arc. D ACB FIG. 56. 24. Prove that the triangle whose sides are x^ + y 1 , # 2 y 2 , %xy is right-angled. 25. AD is an altitude of the triangle ABC ; BD = a; 2 , DC = y 2 , AD = xy ; prove that L BAC = 90. 26. AD is an altitude of AABC, L BAC = 90 ; AD = 4", CD = 3^ ; calculate AB. 27. AD, BC are two vertical poles, D and C being the ends on the ground, which is level; AC = 12', AB = 10', BC = 3'; calculate AD. 28. AD, BC are the parallel sides of the trapezium ABCD ; AB = 6, BC = 9, CD = 5, AD = 14 ; find the area of ABCD. 29. In AABC, AB = AC = 10", BC = 8"; find the radius of the circle which passes through A, B, C. 40 CONCISE GEOMETKY 30. In A ABC, AB*=4", 80 = 5", L ABC = 45; calculate AC. 31. In A ABC, AB*=8", BC = 3", Z. ABC ~60; calculate AC. 32. A regular polygon of n sides is inscribed in a circle, radius r ; its perimeter is p ; prove that its area is ? jir 2 - j- g ) Hence, assuming that the circumference of a circle of radius r is 27ir, prove that the area of the circle is Tir 2 . 33. The slant side of a right circular cone is 10", and the diameter of its base is 8" ; find its height. 34. Find the diagonal of a cube whose edge is 5". 35. A room is 20 feet long, 16 feet wide, 8 feet high; find the length of a diagonal. 36. A piece of wire is bent into three parts AB, BC, CD each of the outer parts being at right angles to the plane containing the other two; AB-12", BC-6", 00 = 12"; find the distance of A from D. 37. A hollow sphere, radius 8", is filled with water until the surface of the water is within 3" of the top. Find the radius of the circle formed by the water-surface. 38. A circular cone is of height h feet, and the radius of its base is r feet ; prove that the radius of its circumscribing sphere is l + feet - 39. A pyramid of height 8" stands on a square base each edge of which is 1'. Find the area of the sides and the length of an edge. 40*. ABCD is a^ rectangle; AB = 6", BC = 8"; it is folded about BD so that the planes of the two parts are at right angles. Find the new distance of A from C. 41. AD is an altitude of the equilateral triangle ABC ; prove that 42. In AABC, Z.ACB90; CD is an altitude; prove AC 2 + BD 2 = BC 2 + AD 2 . 43. ABN, PQN are two perpendicular lines ; prove that PA 2 + QB 8 '44. The diagonals AC, BD of the quadrilateral ABCD are at right angles ; prove that AB 2 + CD 2 = AD 8 + BC 9 . PYTHAGfOKAS' THEOREM 41 45. It m the quadrilateral ABCD, Z ABC = /. ADC = 90 ; prove that AB 2 - AD 2 CD 2 - CB 2 . 46. P is a point inside a rectangle ABCD ; prove that PA 2 + PC 2 = PB 2 + PD 2 . Is this true if P is outside ABCD ? &T In A ABC, L BAG = 90; H, K are the mid-points of AB, AC ; prove that BK 2 + CH 2 = f BC 2 . 48. ABCD is a rhombus ; prove that AC? -l- BD 2 = 2AB 2 + 2BC 2 . 49. In the quadrilateral ABCD, /. ACB = L ADB = 90 ; AH, BK are drawn perpendicular to CD; prove DH 2 + DK 2 = CH 2 + CK 2 . 1)0. PX, PY, PZ, PW are the perpendiculars from a point P to the sides of the rectangle ABCD ; prove that PA 2 -h PB 2 + PC 2 + PD 2 = 2(PX 2 + PY 2 + PZ 2 + PW 2 ). 51. In AABC, Z BAG = 90 and AC-2AB; AC is produced to D so that CD AB; BCPQ is the square on BC; prove BP-BD. 52. AD is an altitude of AABC ; P, Q are points on AD pro- duced such that PD = AB and QD = AC ; prove BQ = CP. 53. In AABC, Z.BAC-90 ; AD is an altitude; prove . BC 54. In AABC, Z.BAC = 90; AX is an altitude; use Fig. 24, page 15, and the proof of Pythagoras' theorem to show that BA 2 = BX . BC ; and deduce that f^ = . ' AC 2 CX 55. In AABC, Z.BAC = 90; AD is an altitude; prove that AD 2 = BD.DC. 56. ABC is an equilateral triangle ; D is a point on BC such that BC = 3BD ; prove AD 2 = \ AB 2 . B?T^ABC is an equilateral triangle; D, E are the mid-points of BC, CD ; prove AE 2 = 13EC 2 . 58. In the AABC, AB = AC = 2BC; BE is an altitude; prove that AE = 7EC. 59. O is any point inside AABC; OP, OQ, OR are the perpen- diculars to BC, CA, AB; prove BP 2 -f CQ 2 + AR 2 = PC 2 + QA 2 +RB 2 . 60. AD is an altitude of AABC ; E is the mid-point of BC ; prove AB 2 - AC 2 = 2BC . DE. 42 CONCISE GEOMETRY 61. Fig. 57 shows a square of side a 4-6 divided up; use area formulae to prove Pythagoras' theorem a 2 4- 6 2 = c 2 . b a FIG. 57. 62*. ABC is a straight line ; ABXY, BCPQ are squares on the same side of AC ; prove PX 2 4- CY 2 = 3(AB 2 4- BC 2 ). 63*. The diagonal AC of the rhombus ABCD is produced to any point P ; prove that PA . PC = PB 2 - AB 2 . 64*. The diagonal AC of the square ABCD is produced to P so that PC = BC ; prove PB 2 = PA . AC. 65*. In AABC, Z.BAC = 90; BCXY, ACPQ, ABRS are squares outside ABC ; prove PX 2 4- RY 2 = 5BC 2 . EXTENSIONS OF PYTHAGOKAS' THEOREM 43 EXTENSIONS OF PYTHAGORAS' THEOREM THEOREM 22 In A ABC, if Z.BAC is obtuse and if CN is the perpendicular to BA produced, then BC 2 = BA 2 + AC 2 + 2BA . AN. FIG. 58. THEOREM 23 In AABC, if L BAG is acute, and if CN is the perpendicular to AB or AB produced, then BC 2 = BA 2 + AC 2 - 2BA . AN. FIG. 59(1). FIG. 59(2). THEOREM 24 In AABC, if AD is a median, then AB 2 + AC 2 = 2 AD 2 + 2DB 2 . A D N C FIG. 60. 44 CONCISE GEOMETRY EXTENSIONS OF PYTHAGORAS' THEOREM EXERCISE VIII 1. Find by calculation which of the following triangles are obtuse-angled, their sides being as follows : (i) 4, 5, 7 ; (ii) 7, 8, 11; (iii) 8,9, 12; (iv) 15, 16,22. 2. Each of the sides of an acute-angled triangle is an exact number of inches; two of them are 12', 15"; what is the greatest length of the third side ? 3. In A ABC, BC = 6, CA-3, AB = 4; CN is an altitude; calculate AN and CN. 4. In AABC, BC = 8, CA=9, AB = 10; CN is an altitude; calculate AN and CN. 5. In AABC, BC = 7, CA=13, AB = 10; CN is an altitude; calculate AN, BN, CN. 6. Find the area of* the triangle whose sides are 9", 10*, 11". 7. ABCDis a parallelogram ; AB = 5", AD= 3"; the projection of AC on AB is 6" ; calculate AC. 8. In AABC, AC8 cms., BC = 6 cms., /.ACB=120; calculate AB. 9. In AABC, AB *= 8 cms., AC = 7, BC = 3 ; prove L ABC = 60. 10. The sides of a triangle are 23, 27, 36; is it obtuse-angled? 11. In AABC, AB = 9*, AC=ir, </ BAC>90; prove BOH". 12. In AABC,AB = 14",BC = 10",CA = 6"; prove L ACB - 1 20. 13. The sides of a A are 4, 7, 9; calculate the length of the shortest median. 14. Find the lengths of the medians of a triangle whose sides are 6, 8, 9 cms. 15. The sides of a parallelogram are '5 cms., 7 cms., and one diagonal is 8 cms. ; find the length of the other. 16. AD is a median of the AABC, AB = 6, AC = 8, AD = 5; calculate BC. 17. In AABC, AB = 4, BC = 5, CA = 8; BC is produced to D so that CD = 5 ; calculate AD. 18. ABC is an equilateral triangle; BC is produced to D so that BC = CD ; prove AD 2 3AB 2 . EXTENSIONS OF PYTIIAGOKAS' THEOREM 45 19. In A ABC, AB = AC; CD is an altitude; prove that BC 2 = 2AB . BD. 20. AB and DC are the parallel sides of the trapezium ABCD ; prove that AC 2 + BD 2 = AD 2 -f BC 2 + 2AB . DC. 21. BE, CF are altitudes of the triangle ABC ; prove that AF.AB-AE.AC. 22. ABCD is a parallelogram; prove that AC 2 -f BD 2 = 2AB 2 + 2BC 2 . 23. ABCD is a rectangle ; P is any point in the same or any other plane ; prove that PA 2 -h PC 2 = PB 2 f PD 2 . 24. In A ABC, AB = AC; AB is produced to D so that AB^BD; prove CD 2 = AB 2 + 2BC 2 . 25. In AABC, D, E are the mid-points of CB, CA; prove that 4(AD* - BE 2 ) = 3(CA 2 - CB 2 ). 26. In AABC, Z.ACB = 90; AB is trisected at P, Q; prove that PC 2 + CQ 2 + QP 2 = AB 2 . 27. The base BC of AABC is trisected at X, Y; prove that AX 2 + AY 2 4- 4XY 2 = AB 2 -h AC 2 . 28. The base BC of AABC is trisected at X, Y; prove that AB 2 - AC 2 = 3( AX 2 - AY 2 ). 29. AD, BE, CF are the medians of AABC; prove that 4(AD 2 + BE 2 -h CF 2 ) = 3(AB 2 + BC 2 + CA 2 ). 30. ABCD is a quadrilateral ; X, Y are the mid-points of AC, BD ; prove that AB 2 -f BC 2 + CD 2 -f DA 2 = AC 2 + BD 2 + 4XY 2 . 31*. ABC is a triangle ; ABPQ, ACXY are squares outside ABC ; prove that BC 2 -h QY 2 = AP 2 + AX 2 . 32*. ABC is a triangle ; D is a point on BC such that p . BD = 0.DC; provethatjp.AB 2 -f^.AC 2 --=(/>4-g)AD 2 H-jp. BD 2 -f q . DC 2 . 33*. AB is a diameter of a circle; PQ is any chord parallel to BA ; O is any point on AB ; prove that OP 2 + OQ 2 = 34* ABCD is a tetrahedron ; L BAC = L CAD = L DAB * 90 ; prove that BCD is an acute-angled triangle. 46 CONCISE GEOMETRY RELATIONS BETWEEN SEGMENTS OF A STRAIGHT LINE EXERCISE IX 1. A straight line AB is bisected at O ; P is any point on AO ; prove PO~J(PB-PA). 2. A straight line AB is bisected at O and produced to P ; prove 3. A straight line AB is bisected at O and produced to P ; prove that PA 2 + PB 2 = 2PO 2 + 2AO 2 . 4. ABCD is a straight line ; X, Y are the mid-points of AB, CD ; prove that AD + BC = 2XY. 5. AB is bisected at O and produced to P ; prove that AO . AP OB.BP + 2AO 2 . 6. AD is trisected at B, C; prove that AD 2 = AB 2 + 2BD 2 . 7. APB is a straight line; prove that AB 2 + AP 2 = 2AB . AP + PB 2 . 8. AB is bisected at C and produced at P; prove that AP 2 = 4AC.CP + BP 2 . 9. ABCD is a straight line ; if AB = CD, prove that AD 2 + BC 2 = 2AB 2 + 2BD 2 . 10. X is a point on AB such that AB.BX = AX 2 ; prove that = 3AX 2 . 11. C is a point on AB such that AB.BC = AC 2 ; prove that AC.BC = AC 2 -BC 2 . 12. X is a point on AB such that AB . BX- AX 2 ; O is the mid- point of AX ; prove that OB 2 = 5 . OA 2 . 13. AB is bisected at O and produced to P so that OB . OP = BP 2 ; prove that PA 2 = 5PB 2 . 14. AB is bisected at C and produced to D so that AD 2 ==3CD 2 ; BC is bisected at P ; prove that PD 2 = 3PB 2 . 15. AB is produced to P so that PA 2 = 4PB 2 + AB 2 ; prove that PA_5 PB~2* RELATIONS BETWEEN SEGMENTS 47 16. ACBD is a straight line such that .= : O is the mid- CB BD point of AB ; prove that (i) DA.DB = DC.DO. (ii) AB.CD-2AD.CB. (iii) OB 2 = OC.OD. (iv)- 1 ^ 1 -^- 2 -. v ' AC AD AB 48 CONCISE GEOMETRY INEQUALITIES THEOREM 26 In the triangle ABC, (i) If AOAB, then /_ ABO (ii) IfZ ABOZ.ACB, then AOAB. a 61. THEOREM 27 If ON is the perpendicular from any point O to a line AB, and if P is any point on AB, then ON < OP. O N P B FIG. 62. THEOREM 28 If ABC is a triangle, BA + AC >BC. INEQUALITIES 49 INEQUALITIES EXERCISE X 1. The bisectors of the angles ABC, ACB of AABC meet at I ; if AB >AC, prove that IB >IC. 2. AD is a median of AABC; if BC<?AD, prove that Z.BAC <90. 3. ABC is an equilateral triangle ; P is any point on BC ; prove AP>BP. 4. In AABC, the bisector of Z.BAC cuts BC at D; prove BA>BD. 5. AD is a median of AABC; if AB>AC, prove that /.BAD <Z.CAD. 6. In AABC, AB AC; BC is produced to any point D; P is any point on AB ; DP cuts AC at Q ; prove AQ > AP. 7. In the quadrilateral ABCD, AD > AB >CD > BC ; prove /. ABC > Z. ADC. Which is the greater, Z. BAD or /. BCD ? 8. ABC is a triangle ; the external bisector of L B AC cuts BC produced at D; prove (i) AB>AC; (ii) CD>AC. 9. ABC is a triangle ; the bisector of /_ BAC cuts BC at D ; if AB>AC, prove BD>DC. 10. ABC is an acute-angled triangle, such that /. ABC = 2 21 ACB ; prove AC<2AB. 11. ABCD is a quadrilateral ; prove that AB + BC + CD >AD. 12. Prove that any side of a triangle is less than half its perimeter. 13. How many triangles can be drawn such that two of the sides are of lengths 4 feet, 7 feet, and such that the third side contains a whole number of feet ? 14. ABC is a A ; D is any point on BC ; prove that AD< (AB + BC + CA). 15. ABCD is a quadrilateral ; AB<BC; /.BAD</.BCD; prove AD>CD. 16. ABC is a A ; P is any point on BC; prove that AP is less than one of the lines AB, AC. 17. O is any point inside the triangle ABC; prove that (i)/_BOC > L BAC ; (ii) BO + OC< BA + AC. 18. A, B are any two points on the same side of CD, A' is the 4 50 CONCISE GEOMETRY image of A in CD ; A'B cuts CD at O ; P is any other point on CD ; prove that AP + PB > AO + OB. 19. AD is a median of AABC; prove AD<|(AB-f AC). 20. O is any point inside AABC ; prove OA + OB -f OC > |(BC -f- CA + AB). 21. In AABC, BC>BA; the perpendicular bisector OP of AC cuts BC at P ; Q is any other point on OP ; prove AQ + QB >AP + PB. 22. Prove that the sum of the diagonals of a quadrilateral is greater than the semiperimcter and less than the perimeter of the quadrilateral. THE INTERCEPT THEOREM 51 THE INTERCEPT THEOREM THEOREM 29 If H, K are the mid-points of the sides AB, AC of the triangle ABC, then (i) HK is parallel to BC. (ii) HK = |BC. A B C FIG. 63. THEOREM 30 If two lines ABCDE, PQRST are cut by the parallel lines BQ, CR, DS so that BC = CD, then QR = RS. :/ \a 7\ z R_ s J \T Fio. 64. 52 CONCISE GEOMETRY THE INTERCEPT THEOREM EXERCISE XI 1. ABC is a A ) H, K are the mid-points of AB, AC ; P is any point on BC ; prove HK bisects AP. 2. In AABC, L BAG = 90 ; D is the mid-point of BC; prove that AD = BC. [From D, drop a perpendicular to AC.] 3. In Fig. 65, if AC = CB and if AP, BQ, CR are parallel, prove 4. In Fig. 66, if AC = CB, and if AP, BQ, CR are parallel, prove that CR = (BQ - AP). " c V B FIG. 66. 5. P. Qi R, S are the mid-points of the sides AB, BC, CD, DA of the quadrilateral ABCD ; prove that PQ is equal and parallel toSR. 6. In AABC, /. ABC = 90; BOX is an equilateral triangle; prove that the line from X parallel to AB bisects AC. 7. ABC is a A ; H, K are the mid-points of AB, AC ; BK, CH are produced to X, Y so that BK KX and CH HY j prove thatXY = 2BC. THE INTERCEPT THEOREM 53 8. O is a fixed point ; P is a variable point on a fixed line AB ; find the locus of the mid-point of OP. 9. O is a fixed point ; P is a variable point on a fixed circle, centre A ; prove that the loo as of the mid-point of OP is a circle whose centre is at the mid-point of OA. 10. Prove that the lines joining the mid points of opposite sides of any quadrilateral bisect each other. 11. If the diagonals of a quadrilateral are equal and cut at right angles, prove that the mid-points of the four sides are the corners of a square. 12. ABCD is a quadrilateral; if AB is parallel to CD, 'prove that the mid-points of AD, BC, AC, BD lie on a straight line. 13. ABC is a A ; AX, AY are the perpendiculars from A to the bisectors of the angles ABC, ACB : prove that XY is parallel to BC. 14. ABCD is a quadrilateral such that BD bisects /.ABC and /.ADB = 90= /.BCD ; AH is the perpendicular from A to BC ; prove AH = 2CD. 15. AD, BE are altitudes of A ABC and intersect at H; P, Q, R are the mid-points of HA, AB, BC ; prove that 16. ABCD is a quadrilateral, having AB parallel to CD; P, Q, R, S are the mid-points of AD, BD, AC, BC; prove that (i) PQ = RS; (ii) PS = (AB + CD) ; (Hi) QR = |(AB~CD). 17 ABC is a A ; D is the mid-point of BC ; P is the foot of the perpendicular from B to the bisector of Z. BAG ; prove that 18. ABC is a A ; D is the mid-point of BC; Q is the foot of the perpendicular from B to the external bisector of /. BAC ; prove that DQ = (AB + AC). 19. ABCD is a quadrilateral having AB=^CD; P, Q, R, S are the mid-points of AD, AC, BD, BC ; prove that PS is perpen- dicular to QR. 54 CONCISE GEOMETRY 20. In Fig. 67, if BD = DC and AP = AQ, prove that BP = CQ and AP = i PIG. 67. aqiKre box ABCD, each edge 13*, rests in the rack of a ^ailwa^tearriage and against the wall : the point of contact E ,is iB^froin the wall ; CE = ED. Prove that C is 5" from the* ^apfj),nd find the distances of A, D from the wall. FIG. 68. 22. ABC is a A ; E, F are the mid-points of AC AB ; BE cuts CF at G; AG is produced to X so that AG GX and cuts BC at D; prove that (i) GBXC is a parallelogram; (ii) DG = |GA = iDA. 23. ABCD is a parallelogram ; XY is any line outside it ; AP, BQ, CR, DS are perpendiculars from A, B, C, D to XY ; prove that AP + CR = BQ + DS. 24* The diagonals AC, BD of the square ABCD intersect at O : THE INTERCEPT THEOREM 55 the bisector of /. BAG cuts BO at X, BC at Y ; prove that" CY=20X. 25*. Two equal circles, centres A, B, intersect at O ; a third equal circle passes through O and cuts the former circles at C, D ; prove that AB is equal and parallel to CD. 26*. A, B are fixed points; P is a variable jx>int ; PAST, PBXY are squares outside APAB ; provo that the mid-point of SX is fixed. [Drop perpendiculars from S, X to AB.] 27*. ABCD is a quadrilateral having AD = BC ; E, F are the mid- points of AB, CD ; prove that EF is equally inclined to A^ and BC. [Complete the parallelogram DABH : at K ; join BK, KF.] RIDERS ON BOOK III SYMMETRICAL PROPERTIES OF A CIRCLE THEOREM 31 AB is a chord of a circle, centre O. (1) If N is the mid-point of AB, then L ON A -90. (2) If ON is the perpendicular from O to AB, then AN = NB. THEOREM 32 AB, CD are chords of a circle, centre O. (1) If AB = CD, then AB and CD are equidistant from O. (2) If AB and CD are equidistant from O, then AB = CD. A corresponding property holds for equal circles. 56 SYMMETRICAL PROPERTIES OF A CIRCLE 57 SYMMETRICAL PROPERTIES OP A CIRCLE EXEEOI8E XII 1. AB is a chord of a circle of radius 10 cms.; AB = 8 cms.; find the distance of the centre of the circle from AB. 2. A chord of length 10 cms. is at a distance of 12 cms. from the centre of the circle ; find the radius. 3. A chord of a circle of radius 7 cms. is at a distance of 4 cms. from the centre ; find ita length. 4. ABC is a A inscribed in a circle; AB = AC = 13", BC==10"; calculate the radius of the circle. 5. In a circle of radius 5 cms., there are two parallel chords of lengths 4 cms., 6 cms. ; find the distance between them. 6. Two parallel chords AB, CD of a circle are 3" apart ; AB = 4", CD = 10" ; calculate the radius of the circle. 7. An equilateral triangle, each side of which is 6 cms., is inscribed in a circle ; find its radius. 8. The perpendicular bisector of a chord AB cuts AB at C and the circle at D ; AB = 6", CD = 1* ; calculate the radius of the circle. 9. ABC is a straight line, such that AB = T, BC = 4"; PBQ is the chord of the circle on AC as diameter, perpendicular to AC; find PQ. 10. P is a point on the diameter AB of a circle ; AP = 2", PB = 8* ; find the length of the shortest chord which passes through P. 11. The centres of two circles of radii 3", 4" are at a distance 5* apart ; find the length of their common chord. 12. Two concentric circles are of radii 3*, 5"; a line PQRS cuts one at P, S and the other at Q, R ; if QR = 2", find PQ. 13. A variable line PQRS cuts two fixed concentric circles of radii a", V at P, S and Q, R; if PQ = #", QR = /, find an equation between x, y, a, &, and prove that PQ.QS is constant. 58 CONCISE GEOMETRY 14. A crescent is formed of two circular arcs of equal radius (see Fig. 71) ; the perpendicular bisector of AB cuts the crescent at C, D ; if CD = 3 cms., AB = 10 cms., find the radii. 15. In Fig. 72, ABCD is the section of a lens; AB = CD = #; BP = PC = y; PQ = z; AB, QP, DC are perpendicular to BC ; calculate in terms of r, y, z the radius of the circular arc AQD. B P C A Fro. 72. 16. AB is a chord of a circle, centre O ; T is any point equidistant from A and B ; prove OT bisects /. ATB. 17. Two circles, centres A, B, intersect at X, Y ; prove that AB bisects XY at right angles. 18. Two circles, centres A, B, intersect at C, D; PCQ is a line parallel to AB cutting the circles at P, Q ; prove PQ = 2AB. 1 9. Two circles, centres A, B, intersect at X, Y ; PQ is a chord of one circle, parallel to XY ; prove AB bisects PQ. 20. A line PQRS cuts two concentric circles at P, S and Q, R ; prove PQ RS. 21. ABC is a triangle inscribed in a circle ; if /. BAC=* 90, prove that the mid-point of BC is the centre of the circle. SYMMETRICAL PROPERTIES OF A CIRCLE 59 22. In Fig. 73, if PQ is parallel to RS, prove PQ = RS. Fre. 73. 23. APB, CPD are intersecting chords of a circle, centre O ; if OP bisects Z.APC, prove AB=-CD. 24. The diagonals of the quadrilateral ABCD meet at O ; circles are drawn through A, O, B ; B, O, C ; C, O, D ; D, O, A ; prove that their four centres are the corners of a parallelogram. 25. AOB, COD are two intersecting chords of a circle ; if AB = CD, prove AO = CO. 26. In Fig. 74, A, C, B are the centres of three unequal circles ; if AC = CB, prove PQ = RS. Fig. 74. 27. AB, CD are two chords of a circle, centre O; if AB>CD, prove O is nearer to AB than to CD. 28. Two circles, centres A, B, intersect at C, D ; PCQ is a line cutting the circles at P, Q ; prove PQ is greatest when it is parallel to AB. 29*. P is any point on a diameter AB of a circle ; QPR is a chord such that L APQ = 45 ; prove that AB 2 = 2PQ 2 + 2PR '. 30*. ABC is a A inscribed in a circle, centre O ; X, Y, Z are the images of O in BC, CA, AB ; prove that AX, BY, CZ bisect each other. 31*. AB, CD are two perpendicular chords of a circle, centre O ; prove that AC 2 + BD 2 - 4OA 2 . 60 CONCISE GEOMETRY ANGLE PEOPERTIES OF A CIRCLE (1) THEOREM 33 If AB is an arc of a circle, centre O, and if P is any point on the remaining part of the circumference, then the angle which arc AB subtends at O equals 2 /. APB, FIG. 75(1). Fio. 75(2). THEOREM 34 (1) If APQB is a circle, L APB= Z.AQB. (2) If AB is a diameter of a circle APB, /. APB = 90. FIG. 76. FIG. 77. ANGLE PROPERTIES OF A CIRCLE (1) THEOREM 35 61 (1) If ABCD is a cyclic quadrilateral, Z.ABC+ /.ADC = 180. (2) If the side AD of the cyclic quadrilateral ABCD is produced to P, = Z.ABC. FIG. 78. 62 CONCISE GEOMETRY ANGLE PROPERTIES OF A CIRCLE (1) EXERCISE XIII 1. ABC is a A inscribed in a circle, centre O; Z.AOC = 130 Z. BOG = 1 50, find L ACB. 2. AB, CD are perpendicular chords of a circle ; Z. BAC = 35, find Z.ABD. 3. ABCD is a quadrilateral such that AB = AC = AD ; if L BAD = 140, find /.BCD. 4. ABCD is a quadrilateral inscribed in a circle ; AB is a diameter; Z ADC -127; find ZBAC. 5. Two chords AB, CD when produced meet at O ; Z. OAD = 31 ; ZAOC-42 ; find ZOBC. 6. Two circles APRB, AQSB intersect at A, B ; PAQ, RBS are straight lines; if ZQPR-=80, ZPRS = 70, find Z.PQS, Z.QSR. 7. P, Q, R are points of a circle, centre O ; L POQ = 54, Z OQR = 36; P, R are on opposite sides of OQ ; find ZQPR and Z.PQR. 8. The diagonals of the cyclic quadrilateral ABCD meet at O ; L BAC -42, ZBOC = 114, ZADB = 33; find /.BCD. 9. ABCD is a cyclic quadrilateral, EABF is a straight line; Z.EAD = 82, ZFBC-74 , ZBDC = 50; find angle be- tween AC, BD. 10. Two chords AB, DC of a circle, centre O, are produced to meet at E ; L AOB - 100, L EBC - 72, L ECB - 84 ; find Z.COD. 11. (i) In Fig. 79, if y = 32, z - 40, find x. (ii) If y + z = 90, prove x = 45. ANGLE PROPERTIES OF A CIRCLE (1) 63 12. D is a point on the baso BC of A ABC ; H, K are the centres of the circles ADB, ADC; if Z. AMD -70, Z.AKD = 80, find Z.BAC. 13. In Fig. 79, if AC cuts BD at O, if y = 20, 2-40, /.BOC- 100, prove L BAG = 2 L BCA. 14. AB, XY arc parallel chords of a circle; AY cuts BX at O; prove OX = OY. 15. Two circles BAPR, BASQ cut at A, B ; PAQ, RAS are straight lines; prove /_ PBR-- Z.QBS. 16. AB is a chord of a circle, centre O ; P is any point on the minor arc AB ; prove /_ AOB + 2 Z. APB - 360. 1 7. ABCD is a cyclic quadrilateral ; if AC bisects the angles at A and C, prove /.ABC = 90. 18. Two lines CAB, OCD cut a circle at A, B and C, D; prove Z.OAD= ZOCB. 19. AB is a diameter of a circle APQRB ; prove Z.APQ+ Z.QRB -270. 20. ABCDEF is a hexagon inscribed in a circle ; prove that /. FAB + Z.BCD+ Z.DEF-360". 21. Two circles ABPR, ABQS cut at A, B ; PBQ, RAS are straight lines ; prove PR is parallel to QS. 22. A, B, C are three points on a circle, centre O ; prove that L BAC = L OBA + L OCA. 23. A, B, C, P are four points on a circle ; prove that a triangle whose sides are parallel to PA, PB, PC is equiangular to AABC. 24. AP, AQ are diameters of the circles APB, AQB ; prove that PBQ is a straight line. 25. OA is a radius of a circle, centre O ; AP is any chord ; prove that the circle on OA as diameter bisects AP. 26. Two chords AOB, COD of a circle intersect at O ; if AO = AC, prove DO = BD. 27. APC is an arc, less than a semicircle, of a circle, centre O; AQOC is another circular arc; prove Z. APC= /.PAQ + 2LPCQ. 64 CONCISE GEOMETRY 28. ABC is a A inscribed in a circle, centre O ; D is the mid- point of BC ; prove /. BOD = Z. BAC. 29. OA, OB, OC are three equal lines; if Z.AOB = 90, prove Z ACB = 45 or 135. 30. Two lines OAB, OCD cut a circle at A, B and C, D; if OB = BD, prove OC = CA. 31. ABCD is a rectangle; any circle through A cuts AB, AC, AD at X, Y, Z ; prove that ABD, XYZ are equiangular triangles. 32. In Fig. 80, O is the centre of the circle; prove Z.AOC + 33. ABCD is a cyclic quadrilateral; AD, BC are produced to meet at E ; AB, DC are produced to meet at F ; the circles EDC, FBC cut at X ; prove EX F is a straight line. 34. AB, CD are perpendicular chords of a circle, centre O ; prove Z.DAB= Z.OAC. 35. In AABC, AB = AC; ABD is an equilateral triangle; prove that L BCD = 30 or 150. 36. ABC is a A ; D is a point on BC ; H, K are the centres of the circles ADB, ADC; if H, D, K, A are concyclic, prove L BAC = 90. 37. ABC is a A ; the bisectors of Z.s ABC, ACB intersect at I and meet AC, AB at P, Q ; if A, Q, I, P are*concyclic, prove L BAC = 60. 38. Two lines EBA, ECD cut a circle ABCD at B, A and C, D ; O is the centre ; prove L AOD - /. BOC = 2 /. BEC. 39. ACB, ADB are two arcs on the same side of AB ; a straight line ACD cuts them at C, D; if the centre of the circle ADB lies on the arc ACB, prove CB = CD. ANGLE PROPERTIES OP A CIRCLE (1) 65 40. ABCD is a quadrilateral inscribed in a circle ; BA, CD when produced meet at E; O is the centre of the circle EAC; prove that BD is perpendicular to OE. 41. ABC is a A inscribed in a circle; AOX, BOY, COZ are three chords intersecting at a |>oint O inside A ABC; prove Z.YXZ= / /BOC- Z.BAC. 42. D is any point on the side AB of A ABC ; points E, F are taken on AC, BC so that Z. EDA = 60 = /. FOB ; a circle is drawn through D, E, F and cuts AB again at G ; prove AEFG is equilateral. 43. ABC is a A ; a line PQR cuts BC produced, CA, AB at P, Q, R; if B, C, Q, R are concyclic, prove the bisectors of Z.8 BPR, BAC are at right angles. 44. APXBYQ is a circle; AB bisects ^PAQ and ^XAY; prove PQ is parallel to XY. 45. ABC is a A ; the bisectors of /.s ABC, ACB meet at I ; the circle BIC cuts AB, AC again at P, Q; prove AB = AQ and AC^AP. 46. AB is a diameter of a circle AQBR; AQ, BR meet when produced at O; use an area formula to prove that BQ.AO-AR.BO. 47. ABC is a A; the bisectors of /.s ABC, ACB intersect at I, and cut AC, AB at Y, Z ; the circles BIZ, CIY meet again at X; prove Z.YXZ+ Z.BIC =180. 48. ABC is a triangle inscribed in a circle; AB = AC; BC is produced to D; AD cuts the circle at E; prove Z.ACE = Z.ADB. 49*. AOB, COD are perpendicular chords of a circle ACBD ; prove that the perpendicular from O to AD bisects, when pro- duced, BC. 50*. ABCD is a quadrilateral inscribed in a circle, centre O ; if AC is perpendicular to BD, prove that the perpendicular from O to AD equals JBC. 51*. OC is a radius perpendicular to a diameter AOB of a circle; P, Q are the feet of the perpendiculars from A, B to any line through C; prove that PC = QB and that AP 2 + BQ* = 2OC 2 . 52*. Two given circles ABP, ABQ intersect at A, B ; a variable 5 66 CONCISE GEOMETRY line PAQ meets them at P, Q ; prove 21 PBQ is of constant size. 53*. ABC is a given A ; P is a variable point on a given circle which passes through B, C; if P, A are on the same side of BC, prove /. PBA - L PCA is constant. 54*. In Fig. 81, the circles are given; prove Z.PRQ is of constant size. P FIG. 81. 55*. AB is a fixed chord, and AP a variable chord of a given circle ; C, Q are the mid-points of AB, AP ; prove Z. AQC has one of two constant values. 56*. A variable circle passes through a fixed point A and cuts two given parallel lines at P, Q such that /.PAQ = 90; prove that the circle passes through a second fixed point. 57*. Two circles PRQ, PSQ intersect at P, Q; the centre O of circle PRQ lies on circle PSQ; the diameter PS of circle PSQ cuts circle PRQ at R ; prove QR is parallel to OP, ANGLE PROPERTIES OF A CIRCLE (2) 67 ANGLE PROPERTIES OF A CIRCLE (2) THEOREM 40 If P is any point on a circle, centre O, and if PX is the tangent at P, then Z. OPX = 90. P,Q FIG. 82. THEOREM 41 If PA is any chord of a circle PKA, and if PX is the tangent at P, K and X being on opposite sides of PA, then Z.APX= L AKP. FIG. 83. 68 CONCISE GEOMETRY ANGLE PROPERTIES OP A CIRCLE (2) EXERCISE XIV 1. A line TBC cuts a circle ABC at B, C; TA is a tangent; if L TAG = 118, Z.ATC = 26, find Z.ABC. 2. ABC is a minor arc of a circle ; the tangents at A, C meet at T; if Z.ATC = 54, find /.ABC. 3. AOC, BOD are chords of a circle ABCD ; the tangent at A meets DB produced at T ; if Z.ATD = 24, /.COD = 82, Z. TBC = 146, find ZL BAG. Find also the angle between BD and the tangent at C. 4. The aides BC, CA, AB of a A touch a circle at X, Y, Z ; ,/ ABC = 64, Z. ACB = 52; find Z.XYZ, ,/XZY. 5. Three of the angles of a quadrilateral circumscribing a circle are 70, 84, 96 in order; find the angles of the quadri- lateral whose vertices are the points of contact. 6. TBP, TCQ are tangents to the circle ABC; Z.PBA = 146, Z.QCA = 128; find /.BAG and ZLBTC. 7. In A ABC, L ABC = 50, Z. ACB = 70; a circle touches BC, AC produced, AB produced at X, Y, Z ; find Z. YXZ. 8. A chord AB of a circle is produced to T ; TC is a tangent from T to the circle; prove Z.TBC = /.ACT. 9. Two circles APB, AQB intersect at A, B ; AP, AQ are the tangents at A, prove Z.ABP = Z.ABQ. 10. DA is the tangent at A to the circle ABC ; if DB is parallel to AC, prove Z. ADB => Z. ABC. 11. In A ABC, AB = AC; D is the mid-point of BC; prove that the tangent at D to the circle ADC is perpendicular to AB. 12. BC, AD are parallel chords of the circle ABCD ; the tangent at A cuts CB produced at P ; PD cuts the circle at Q ; prove Z.PAQ=Z.BPQ. 13. Two circles ACB, ADB intersect at A, B ; CA, DB are tangents to circles ADB, ACB at A, B ; prove AD is parallel to BC. 14. CA, CB are equal chords of a circle ; the tangent ADE at A meets BC produced at D ; prove Z. BDE = 3 Z. CAD. 15. The bisector of Z.BAC meets BC at D ; a circle is drawn ANGLE PROPERTIES OF A CIRCLE (2) 69 touching BC at D and passing through A ; if it cuts AB, AC at P, Q, prove ^PDB= Z.QDC. 16. Two circles APB, AQB intersect at A, B; AQ, AP are the tangents at A ; if PBQ is a straight line, prove Z. PAQ = 90. 17. ABCD is a quadrilateral inscribed in a circle ; the tangents at A, C meet at T ; prove Z. ATC = Z. ABC ~ L ADC. 18. Two circles intersect at A, B; the tangents at B meet the circles at P, Q ; if Z. PBQ is acute, prove Z. PAQ = 2 Z. PBQ. What happens if Z. PBQ is obtuse ? 19. ABC is a A inscribed in a circle ; the tangent at C meets AB produced at T ; the bisector of L ACB cuts AB at D ; prove TC = TD. 20. AOB is a diameter of a circle, centre O; the tangent at B meets any chord AP at T ; prove Z.ATB = Z.OPB. 21. ABCDE is a pentagon inscribed in a circle ; AT is the tangent at A, T and D being on opposite sides of AB ; prove L BCD + ZlAED=:lS0 -f ,/BAT. 22. In A ABC, AB = AC; a circle is drawn to touch BC at B and to pass through A ; if it cuts AC at D, prove BC == BD. 23. In AABC, Z. BAC = 90 ; D is any point on BC ; DP, DQ are tangents at D to the circles ABD, ACD ; prove /_ PDQ = 90. 24. AB is a diameter of a circle ABC ; TC is the tangent from a point T on AB produced ; TD is drawn perpendicular to TA and meets AC produced at D ; prove TC = TD. 25. EAF, CBD are tangents at the extremities of a chord AB of a circle, E and C being on the same side of AB ; if AB bisects Z. CAD, prove Z. EAC = L ADC. 26. Two circles touch internally at A ; the tangent at any point P on the inner cuts the outer at Q, R ; AQ, AR cut the inner at H, K; prove AS PQH, APK are equiangular. 27. PQ is a common tangent to two circles CDP, CDQ; prove that Z. PCQ + Z. PDQ = 180. 28. Two chords AOB, COD of a circle cut at O ; the tangents at A, C meet at X ; the tangents at B, D meet at Y ; prove Z.AXC+ ^BYD==2Z.AOD. 29. I is the centre of a circle touching the sides of AABC; a larger concentric circle is drawn ; prove that it cuts off equal portions from AB, BC, CA. 70 CONCISE GEOMETRY 30. PQ, PR are equal chords of a circle ; PQ and the tangent at R intersect at T ; prove L PRQ = 60 \ L PTR. 31. The diameter AB of a circle, centre O, is produced to T so that OB = BT ; TP is a tangent to the circle ; prove TP = PA. 32. The bisector of L BAG cuts BC at D ; a circle is drawn through D and to touch AC at A ; prove that its centre lies on the perpendicular from D to AB. 33. Three circles, centres A, B, C, have a common point of inter- section O ; also their common chords are equal ; prove that O is the centre of the circle inscribed in AABC. 34. AB is a chord of a circle \ the tangents at A, B meet at T ; AP is drawn perpendicular to AB, and TP is drawn per- pendicular to TA ; prove that PT equals the radius. 35. Two circles ABD, ACE intersect at A ; BAC, DAE are straight lines ; prove that the angle between DB and CE equals the angle between the tangents at A. 36. Assuming the result of ex. 21 (page 63), what special cases can be obtained by taking (i) Q very close to S, (ii) Q very close to B, (iii) A very close to B ? 37. A, B are given points on a given circle ; P is a variable point on the circle ; the circles whose diameters are AB and AP intersect at Q. Find the position of Q when P is very close to B. 38. OA is a chord of a circle, centre C ; T is a point on the tangent at O such that OA = OT and L AOT is acute ; TA is pro- duced to cut OC at B ; prove that /. OBA = J / OCA, Find the position of B when A is very close to O. PROPERTIES OF EQUAL ARCS AND CIRCLES 71 PROPERTIES OF EQUAL ARCS AND EQUAL CIRCLES THEOREM 37. H, K are the centres of two equal circles APB, CQD. (i) IfZ.AHB= zlCKD, then arc AB = arc CD. (ii) If /_APB=* Z.CQD, then arc AB = arc CD. THEOREM 38. H ( K are the centres of two equal circles APB, CQD. If arc AB = arc CD, then (i) L AHB = L CKD, and (ii)/.APB = Z.CQD. THEOREM 39. APB, CQD are two equal circles. (i) If chord AB = chord CD, then arc AB = arc CD. (ii) If arc AB = arc CD, then chord AB = chord CD. These properties also hold for equal arcs in the same circle. 72 CONCISE GEOMETRY PROPERTIES OF EQUAL ARCS AND EQUAL CIRCLES EXERCISE XV 1. ABCD Ls a square and AEF is an equilateral triangle inscribed in the same circle ; calculate the angles of AECD. 2. AB is a side of a regular hexagon and AC of a regular octagon inscribed in the same circle ; calculate the angles of AABC. 3. ABCD is a quadrilateral inscribed in a circle ; AC cuts BD at O : DA, CB when produced meet at E ; AB, DC when pro- duced meet at F ; if Z. AEB = 55, Z. BFC = 35, Z. DOC = 85, prove arc BC = twice arc AB. 4. ABC is a triangle inscribed in a circle : the tangent at A meets BC produced at T; /.BAT ^135,ZATB- 30 ; find the ratio of the arcs AB and AC. 5. A, B are two points on the circle ABCD such that the minor arc AB is half the major arc AB ; Z. DAB = 74 ; arc BC = arc CD; calculate/. A BD andZ_BDC. 6. ABCD is a quadrilateral inscribed in a circle; Z.ADB = 25, Z. DBC = 65 ; prove arc AB + arc CD = arc BC + arc AD. 7. AB, CD are parallel chords of a circle ; prove arc AD arc BC. 8. ABCD is a cyclic quadrilateral ; if AB = CD, prove Z. ABC = </ BCD. 9. A circle AOBP passes through the centre O of a circle ABQ ; prove that OP bisects Z.APB. 10. ABP, ABQ are two equal circles; PBQ is a straight line; prove AP = AQ. 11. AB, BC, CD are equal chords of a circle, centre O ; prove that AC cuts BD at an angle equal to Z. AOB. 12. ABCD is a square and APQ an equilateral triangle inscribed in the same circle, P being between B and C; prove arc BP = arcPC. 13. On a clock-face, prove that the line joining 4 and 7 is perpendicular to the line joining 5 and 12. PROPERTIES OF EQUAL ARCS AND CIRCLES 73 14. X, Y are the mid-points of the arcs AB, AC of a circle; XY cuts AB, AC at H, K ; prove AH = AK. 15. APB, AQB are two equal circles; AP is a tangent to the circle AQB, prove AB = BP. 16. ABCD is a rectangle inscribed in a circle; DP is a chord equal to DC ; prove PB = AD. 17. A hexagon is inscribed in a circle; if two pairs of opposite sides are parallel, prove that the third pair is also parallel. 18. ABC is a A inscribed in a circle; any circle through B, C cuts AB, AC again at P, Q ; BQ, CP are produced to meet tho circle ABC at R, S ; prove AR AS. 19. ABCDEF is a hexagon inscribed in a circle; if /.ABC = Z.DEF, prove AF is parallel to CD. "20. CD is a quadrant of the circle ACDB ; AB is a diameter ; AD cuts BC at P ; prove AC = CP. 21. ABC is a A inscribed in a circle, centre O ; P is any point on the side BC ; prove that the circles OBP, OCP are equal. 22. In AABC, AB = AC; BC is produced to D; prove that the circles ABD, ACD are equal. 23. ABCD is a quadrilateral inscribed in a circle ; CD is produced to F; the bisector of Z.ABC cuts the circle at E; prove that DE bisects /.ADF. 21. ABCD is a cyclic quadrilateral ; BC and AD are produced to meet at E ; a circle is drawn through A, C, E and cuts AB, CD again at P, Q ; prove PE = EQ. 25. AB, AC are equal chords of a circle ; BC is produced to D so that CD = CA; DA cuts the circle at E; prove that BE bisects /.ABC. 26. ABC is an equilateral triangle inscribed in a circle ; H, K are the mid-points of the arcs AB, AC; prove that HK is trisected by AB, AC. 27. AB, BC are two chords of a circle (AB>BC) ; the minor arc AB is folded over about the chord AB and cuts AC at D ; prove BD = BC. 28. ABCD is a quadrilateral inscribed in a circle ; X, Y, Z, W are the mid-points of the arcs AB, BC, CD, DA; prove that XZ is perpendicular to YW. 29. In AABC, AB>AC; the bisectors of ^ ABC, ACB meet 74 CONCISE GEOMETRY at I ; the circle BIG cuts AB, AC at P, Q ; prove PI =r 1C and QI = IB. 30. ABC is a triangle inscribed in a circle, centre O ; PQ is the diameter perpendicular to BC, P and A being on the same side of BC ; prove /. ABC ~ L ACB = /. POA. 31*. In Fig. 85, the circles are equal and AD = BC; prove XBYD is a parallelogram. FIG. 85. 32*. In AABC, AB = AC ; D is any point on BC; X, Y are the centres of the circles ABD, ACD; XP, YQ are the per- pendiculars to AB, AC; prove XP = YQ. 33*. AB, CD are two perpendicular chords of a circle, centre O ; prove that AC 2 + BD 2 = 4OA 2 . [Use Theorem 25(2).] 34*. A 1 A 2 A 3 . . . . A 2rt is a regular polygon of 2n sides ; if 2i&> p>q>r>8, prove that A p A r f is perpendicular to A g A 5 if p + r = q + s + n. 35*. ABC is an equilateral triangle inscribed in a circle ; D, E are points on the arcs AB, BC such that AD = BE, prove AD + DB = AE. 36*. C is the mid-point of a chord AB of a circle ; D, E are points on the circle on opposite sides of AB such that L DAC = Z.AEC; prove that Z.ADC= Z.EAC. 37*. P, Q, R are points on the sides BC, CA, AB of AABC such that ^PQR=ZABC and Z.PRQ=Z.ACB; prove that the circles AQR, BRP, CPQ meet at a common point, K say, and are equal; prove also that (i) /.BKC = 2z.BAC; (ii) AK=*BK = CK; (iii) PK is perpendicular to QR. 38*. Two fixed circles cut at A, B ; P is a variable point on one ; PROPERTIES OF EQUAL ARCS AND CIRCLES 75 PA, PB when produced cut the other at QR ; prove QR is of constant length. 39*. A is a fixed point on a fixed circle ; B is a fixed point on a fixed line BC ; a variable circle through A, B cuts BC at P and the fixed circle at Q; prove that PQ cuis the fixed circle at a fixed point. 76 CONCISE GEOMETEY L-ENGTHS OF TANGENTS AND CONTACT OF CHICLES THEOREM 42 If TP, TQ are the tangents from T to a circle, centre O, then (i) TP - TQ. (ii) Z.TOP = TOQ. (in) OT bisects Z. PTQ. P T FIG. 86. THEOREM 43 If two circles, centres A, B, touch, internally or externally, at P, then APB is a straight line. X Y FIG. 87(1) FIG. 87(2). If the circles touch externally (Fig. 87(1)), the distance between the centres AB = sum of radii. If the circles touch internally (Fig. 87(2)), the distance between the centres AB = difference of radii. LENGTHS OF TANGENTS 77 LENGTHS OP TANGENTS AND CONTACT Otf CIRCLES EXEBCISE XVI 1. A circle, radius 5 cms., touches two concentric circles and encloses the smaller: the radius of the larger circle is 7 cms. : what is the radius of the smaller 1 2. Three circles, centres A, B, C, touch each other externally; AB = 4", BC - 6", CA - V ; find their radii.} In AABC, AB = 4", BC = 7", CA = 5* ; two circles with B, C as centres touch each other externally ; a circle with A as centre touches the others internally ; find their radii. Fig. 88 is formed of three circular arcs of radii 6*7 cms., 2-2 cms., 3*1 cms.; X, Y, Z are the centres of the circles; find the lengths of the sides of 3. 4. FIG. 88. 5. In Fig. 89, AB is a quadrant touching AD at A and the quadrant BC at B ; L ADC - 90, AD - 12", DC = 9" ; find the radii of the circles. ^ FIG. 89. 6. The distance between the centres of two circles of radii 4 cms., 7 cms. is 15 cms.; what is the radius of the least circle that can be drawn to touch them and enclose the smaller circle 1 78 CONCISE GEOMETRY 7. C is a point on AB such that AC = 5", CB = 3* ; calculate the radius of the circle which touches AB at C and also touches the circle on AB as diameter. 8. A, B aje the centres of two circles of radii 5 cms., 3 cms. ; AB 12 cms.; BC is a radius perpendicular to BA; find the radius of a circle which touches the larger circle and touches the smaller circle at C. [Two answers.] 9. AB, BC are two equal quadrants touching at B ; their radii are 12 cms.; find the radius of the circle which touches arc AB, arc BC, AC. B FIG. 90. 10. In A ABC, AB = 4", 80 = 6", CA = 7"; a circle touches BC, CA, AB at X, Y, Z ; find BX and AY. 11. In AABC, AB = 3", BC = 7*, CA = 9*; a circle touches CA produced, CB produced, AB at Q, P, R; find AQ, BR. 12. Two circles of radii 3 cms., 12 cms. touch each other externally ; find the length of their common tangent. 13. The distance between the centres of two circles of radii 11 cms., 5 cms. is 20 cms. ; find the lengths of their exterior and interior common tangents. 14. The distance between the centres of two circles is 10 cms., and the lengths of their exterior and interior common tangents are 8 cms., 6 cms. ; find their radii. 15. ABCD is a square of side 7*; C is the centre of a circle of radius 3* ; find the radius of the circle which touches this circle and touches AB at A. 16. In one corner of a square frame, side 3', is placed a disc of radius 1' touching both sides ; find the radius of the largest disc which will fit into the opposite corner. 17. a, b are the lengths of the diameters of two circles which touch each other externally ; t is the length of their common tangent ; prove that t 2 = ab. 18. Two circles of radii 4 cms., 9 cms. touch each other externally ; LENGTHS OF TANGENTS 79 find the radius of the circle which touches each of these circles and also thejr common tangent. [Two answers : use ex. 17,] 19. OA = a*, OB = ^, Z_AOB = 90; two variable circles are drawn touching each other externally, one of them touches OA at A, and the other touches OB at B ; if their radii are af 9 y", prove that (x -f a) (y f b) is constant. If a = 8, b = 6, x - 4, calculate y. 20. Four equal spheres, each of radius 1*, are fixed in contact with each other on a horizontal table, with their centres at the corners of a square; a fifth equal sphere rests on them ; find the height of its centre above the table, 21. A circle touches the sides of A ABC at X, Y, Z; if Y, Z are the mid-points of AB, AC, prove that X is the mid-point of BC. 22. Two circles touch each other at A ; any line through A cuts the circles at P, Q; prove that the tangents at P, Q are parallel. 23. ABCD is a quadrilateral circumscribing a circle, prove that AB-f CD = BC + AD. 24. ABCD is a parallelogram; if the circles on AB and CD as diameters touch each other, prove that ABCD is a rhombus. 25. Two circles touch externally at A ; PQ is their common tangent ; prove that the tangent at A bisects PQ and that Z. PAQ = 90. 26. In Fig. 91, prove AB - CD = BC - AD. 80 CONCISE GEOMETRY 27. ABCDEF is a hexagon circumscribing a circle; prove that AB + CD + EF - BC + DE + FA. 28. In A ABC, L BAC = 90 ; O is the mid-point of BC ; circles are drawn with AB and AC as diameters ; prove that two circles can be drawn with O as centre to touch each of these circles. 29. Two circles touch externally at A ; AB is a diameter of one ; BP is a tangent to the other ; prove that /. APB = 45 IZ.ABP. 30. A BCD is a quadrilateral circumscribing a circle, centre O ; prove Z.AOB+ Z. COD -180. 31. Two circles touch internally at A ; a chord PQ of one touches the other at R ; provo L PAR = Z. QAR. 32. Two circles touch internally at A ; any line PQRS cuts one at P, S and the other at Q, R ; prove /_ PAQ = Z.RAS. 33. Two equal circles, centres X, Y, touch at A ; P, Q aro points, one on each circle such that /_ PAQ = 90 ; prove that PQ is parallel to XY. 34. Two circles touching internally at A ; P, Q are points, oue on each circle, such that Z PAQ ^=90; prove that the tangents at P and Q are parallel. 35. Two circles touch at A ; any line PAQ cuts one circle at P, and the other at Q ; prove that the tangent at P is perpendicular to the diameter through Q. 36. In A ABC, Z. ABC -90; a circle, centre X, is drawn to touch AB produced, AC produced, and BC; prove Z.AXC-45 . 37. Two circles touch externally at A ; a tangent to one of them at P cuts the other circle at Q, R; prove Z.PAQ+ Z.PAR = 180. 38. Two circles, centres A, B, touch externally at P ; a third circle, centre C, encloses both, touching the first at Q and the second at R ; prove /. BAC = 2 /. PRQ. 39. A circle, centre A, touches externally two circles, centres B, C at X, Y ; XY cuts the circle, centre C, at Z ; prove BX is parallel to CZ. 40. PR, QR are two circular arcs touching each other at R, and LENGTHS OF TANGENTS 81 touching the unequal lines OP, OQ at P, Q; prove that Z. PRQ = 180 - \ L POQ (see Fig. 92). FIG. 02. 41*. A circle PBQ, centre A, passes through the centre B of a circle RST; if RP, SQ are common tangents, prove that PQ touches the circle RST. 42*. O is the centre of a fixed circle ; two variable circles, centres P, Q, touch the fixed circle internally and each other externally; prove that the perimeter of AOPQ is constant. 43*. Two given circles touch internally at A ; a variable line through A cuts the circles at P, Q ; prove that the j)er- pendicular bisector of PQ passes through a fixed point. 44*. OA, OB are two radii of a circle, such that /. AOB = 60 ; a circle touches OA, OB and the arc AB ; prove that its radius = JOA. 45*. C is the mid-point of AB ; semicircles are drawn with AC, CB, AB as diameters and on the same side of AB ; a circle is drawn to touch the three semicircles; prove that its radius = JAC. 46*. A square ABOD is inscribed in a circle, and another square PQRS is inscribed in the minor segment AB ; prove that AB = 5PQ. 82 CONCISE GEOMETRY CONVERSE PROPERTIES THEOREM 36 (i) If Z.APB= ZAQB and if P, Q are on the same side of AB, the four points A, B, P, Q lie on a circle, (ii) If /.APB + Z AQB- 180 and if P, Q are on opposite sides of AB, the four points A, B, P, Q lie on a circle, (iii) If . APB -= 90, then P lies on the circle whose diameter is AB. CONVERSE OF THEOREM 41 If C and T are points on opposite sides of a line AB and such that /_ BAT = ACB, then AT is a tangent to the circle which passes through A, C, B. FIG. 95 ler at CONVERSE PROPERTIES 83 CONVERSE PROPERTIES EXERCISE XVII 1. ABCD is a parallelogram; if /.ABC ^60, prove that the centre of the circle ABD lies on the ciicle CBD. 2. BE, CF are altitudes of A ABC; prove that ^AEF-= ZABC. 3. The altitudes AD, BE of AABC intersect at H ; prove that Z_DHC= ZDEC. 4. ABCD is a parallelogram : any circle through A, D cuts AB, DC at P, Q ; prove that B, C, Q, P are concyclic. 5. ABC is a A inscribed in a circle ; BE, CF are altitudes of AABC ; prove that EF is parallel to the tangent at A. 6. The circle BCGF lies inside the circle ADHE; OABCD and OEFGH are two lines cutting them ; if A, B, F, E are con- cyclic, prove that C, D, H, G are concyclic. 7. ABCD is a parallelogram ; AC cuts BD at O ; prove that the circles AOB, COD touch each other. 8. A line AD is trisected at B, C ; BPC is an equilateral triangle ; prove that AP touches the circle PBD. 9 AB is a diameter, AP and AQ are two chords of a circle ; AP, AQ cut the tangent at B in X, Y ; prove that P, X, Y, Q are concyclic. 10. ABC is a A inscribed in a circle ; any lino parallel to AC cuts BC at X, and the tangent at A at Y ; prove B, X, A, Y are concyclic. 11. In Fig. 96, BQP and BAC are equiangular isosceles triangles; prove that QA is parallel to BC. A 12. ABCD is a parallelogram ; a circle is drawn touching AD at A and cutting AB, AC at P, Q; prove that P, Q, C, Bare concyclic. 84 CONCISE GEOMETRY 13. ABCD is a rectangle; the line through C perpendicular to AC cuts AB, AD produced at P, Q ; prove that P, D, B, Q are concyclic. 14. In A ABC, L BAG = 90 ; the perpendicular bisector of BC cuts CA, BA produced at P, Q ; prove that BC touches the circle CPQ. 15. ABCDE is a regular pentagon ; BD cuts CE at O ; prove that BC touches the circle BOE. 16. OY is the bisector of ZLXOZ; P is any point; PX, PY, PZ are the perpendiculars to OX, OY, OZ ; prove that XY-YZ. 17. AA 1 , BB 1 , CC 1 are equal arcs of a circle ; AB cuts A 1 B 1 at P ; AC cuts A 1 C 1 at Q ; prove that A, A 1 , P, Q are concyclic. 18. CA, CB are two fixed radii of a circle ; P is a variable point on the circumference ; PQ, PR are the perpendiculars from P to CA, CB ; prove that QR is of constant length. 19. ABC is a A inscribed in a circle; a line parallel to AC cuts BC at P, and the tangent at A at T ; prove that 20. O is a fixed point inside a given A ABC ; X is a variable point on BC ; the circles BXO, CXO cut AB, AC at Z, Y ; prove that (1) O, Y, A, Z are concyclic, (2) the angles of AXYZ are of constant size. 21. Four circular coins of unequal sizes lie on a table so that each touches two, and only two, of the others; prove that the four points of contact are concyclic. 22. ABC, ABD are two equal circles ; if AB = BC, prove that AC touches the circle ABD. 23. AB, CD are two intersecting chords of a circle ; AP, CQ are the perpendiculars from A, C to CD, AB ; prove that PQ is parallel to BD. 24. Prove that the quadrilateral formed by the external bisectors of any quadrilateral is cyclic. 25. AC, BD are two perpendicular chords of a circle ; prove that the tangents at A, B, C, D form a cyclic quadrilateral. 26. AB, AC are two equal chords of a circle ; AP, AQ are two chords cutting BC at X, Y ; prove P, Q, X, Y are concyclic. 27. The diagonals of a cyclic quadrilateral ABCD intersect at CONVEKSE PBOPERTIES 85 right angles at O ; prove that the feet of the perpendiculars from O to AB, BC, CD, DA are concyclic. 28. AOB, COD are two perpendicular chords of a circle ; DE is any other chord ; AF is the perpendicular from A to DE ; prove that OF is parallel to BE. 29. ABC is a A inscribed in a circle ; AD is an altitude of AABC ; DP is drawn parallel to AB and meets the tangent at A at P ; prove Z.CPA = 90. 30. BE, CF are altitudes of AABC ; X is tho mid-point of BC ; prove that XE-XF. 31. BE, CF are altitudes of AABC; X is the mid-point of BC; prove that Z. FXE - 180 - 2 L BAG. 32. Two circles APRB, ASQB intersect at A, B ; PAQ and RAS are straight lines ; RP and QS are produced to meet at O ; prove that O, P, B, Q are concyclic. 33. AOB, COD are two perpendicular diameters of a circle ; two chords CP, CQ cut AB at H, K ; prove that H, K, Q, P are concyclic. 34. The side CD of the square ABCD is produced to E ; P is any point on CD ; the line from P perpendicular to PB cuts the bisector of /. ADE at Q ; prove BP= PQ. 35*. AB, CD are parallel chords of a circle, centre O ; CA, DB are produced to meet at P ; the tangents at A, D meet at T ; prove that A, D, P, O, T are concyclic. 36*. X, Y are the centres of the circles ABP, ABQ ; PAQ is a straight line ; PX and QY are produced to meet at R ; prove that X, Y, B, R are concyclic. 37*. BE, CF are altitudes of AABC; Z is the mid-point of AB ; prove that Z.ZEF= /. ABC /. BAC. 38*. PQ is a chord of a circle ; the tangents at P, Q meet at T ; R is any point such that TR -= TP ; RP, RQ cut the circle again at E, F ; prove that EF is a diameter. 39*. PQ, CD are parallel chords of a circle ; the tangent at D cuts PQ at T ; B is the point of contact of the other tangent from T ; prove that BC bisects PQ. 40*. ABCD is a parallelogram ; O is a point inside ABCD such that L AOB + Z. COD = 180; prove that Z.OBC = Z.ODC. 86 CONCISE OKOMETRY MENSURATION 1. For a circle of radius r inches, (i) the length of the circumference^- 2?rr in. (li) the area of the cucle ~TTI* sq. in. (in) the length of an arc, which subtends 6 at tho centre of t\ the chcle, o,.:: x ^irr in. A (iv) tho area of a sector of a circle of angle = - x -n-r 2 aq. in. 2. For a sphere of radius r inches, (i) the area of surface of sphere = 47rr 2 sq. in. (ii) tlie volume of the sphere \Trr* cub. in. (in) the area of the surface intercepted between two parallel planes at distance d inches apart %-irrd s<{ in 'A For a circular cylinder, radius r inches, height h inches, (i) tho area of the curved surface - 2?rrA s<j. in. (ii) the volume of the cylinder irr^h cub. in. 4. For a circular ume, radius of base r inches, height h inches, length of slant edge I inches, (i) P~*a + /A (ii) area of the curved surface = ?rrZ sq. in. (Hi) volume of cone = ^7rr% cub. in. 5. (i) The volume of any cylinder = area of base x height. (ii) The volume of any pyramid = \ area of base x height. w= 2 7 2 approx. or 3*1416 approx. MENSURATION 87 MENSURATION EXERCISE XVIII 1. Find (1) the circumference, (2) the area of a circle of radius (i) 4", (ii) 100 yards. 2. The circumference of a circle is 5 inches ; what if its radius correct to ^ inch ? 3. The area of a circle is 4 sq. cms. ; what is its radius correct to iV cm. 1 4. An arc of a circle of radius 3 inches subtends an angle of 40 at the centre ; what is its length correct to T \> inch 1 5. The angle of a sector of a circle is 108, and its radius is 2*5 cms. ; what is its area ? 6. A square ABCD is inscribed in a circle of radius 4 inches ; what is the area of the minor segment cut off by AB. 7. AB is an arc of a circle, centre O ; AO = 5 cms. and arc AB = 5 cms. ; find /. AOB, correct to nearest minute. 8. A piece of flexible wire is in the form of an arc of a circle of radius 4 '8 cms. and subtends an angle of 240 at the centre of the circle : it is bent into a complete circle : what is the radius 1 9. A horse is tethered by a rope 5 yards long to a ring which can slide along a low straight rail 8 yards long ; what is the area over which the horse can graze 1 10. OA, OB are two radii of a circle ; prove that the area of sector AOB equals |OA x arc AB. 11. What is the area contained between two concentric circles of radii 6 inches, 3 inches 1 A C FIG. 97. 12. In Fig. 97, AB, BC, CD, DA are quadrants of equal circles of radii B cms., touching each other. Find the area of the figure. 88 CONCISE GEOMETRY 13. Find (i) the volume, (ii) the total surface of a closed cylinder, height 8*, radius 5". 14. 1 Ib. of tobacco is packed in a cylindrical tin of diameter 4* and height 8" ; what would be the height of a tin of diameter 3" which would hold Ib. of tobacco, similarly packed 1 15. How many cylindrical glasses 1" in diameter can be filled to a depth of 3" from a cylindrical jug of diameter 5" and height 12"! 16. Find (i) the volume, (ii) the area of the curved surface of a circular cone, radius of base 5*, height 12". 17. A sector of a circle of radius 5 cms. and angle 60 is bent to form the surface of a cone; find the radius of its base. 18. The curved surface of a circular cone, height 3*, radius of base 4* is folded out flat. What is the angle of the sector so obtained ] 19. Find (i) the volume, (ii) the total area of the surface of a pyramid, whose base is a square of side 6* and whose height 184". 20. Find (i) the volume, (ii) the area of the surface of a sphere of diameter 5 cms. 21. Taking the radius of the earth as 4000 miles, find the area between latitudes 30 N and 30 S. What fraction is this area of the area of the total surface of the earth ? 22. Two cylinders, diameters 8* and 6", are filled with water to depths 10*, 5" respectively: they are connected at the bottom by a tube with a tap : when the tap is turned on, what is the resulting depth in each cylinder 1 23. Three draughts, 1" in diameter, are placed flat on a table and an elastic band is put round them. Find its stretched length. 24. What is the length of a belt which'passes round two wheels of MENSURATION 89 diameters 2", 4", so that the two straight portions cross at right angles 1 (see Fig. 98). FIG. 93. 25. A circular metal disc, 9" in diameter, weighs 6 Ib. ; what is the weight of a disc of the same metal, 6" in diameter and of the same thickness 1 26. Find the volume of the greatest circular cylinder that can be cut from a rectangular block whose edges are 4", 5", 6". 27. Fig. 99 (not drawn to scale) is a street plan, in which EF is a quadrant and the angles at A, H, D, E, F are 90; AE = AB = DF = 100 yards; HD = 300 yards; CH = 150 yards. Find the two distances of A from D by the routes (i) AEFD, (ii)-ABCD. Find also the area in acres of the plot ABCOFE. D C H 28. AB, BC, CA are three circular arcs, each of radius 6 cms. and touching each other at A, B, C (see Fig. 100) (i) Calculate the area of the figure, (ii) Find its perimeter. A FIG. iOO. 90 CONCISE GEOMETRY 29. Draw a circle of radius 5 cms. and place in it a chord AB of length 4 cms. ; find the area of the major segment AB, making any measurements you like. 30. A rectangular lawn 15 yards by 10 yards is surrounded by flower-beds : a man can, without stepping off the lawn, water the ground within a distance of 5 feet from the edge. What is the total area of the beds he can so water 1 What would be the area within his reach, if the lawn was in the shape of (i) a scalene triangle, (ii) any convex polygon, of perimeter 50 yards 1 31. ABC is a right-angled triangle ; circles are drawn with AB, BC, CA as diameters ; prove that the area of the largest is equal to the sum of the areas of the other two circles. 32. Fig. 101 represents four semicircles; AC = DB and XOV bisects AB at right angles. Prove that (i) Curves AXB, AVB are of equal lengths ; (ii) Area of figure area of circle on XV as diameter. 33. In Fig. 102, BQA, ARC, BSARC are semicircles, prove that the sum of the areas of the lunes BSAQ, CRAP equals the area of A ABC. MENSURATION 91 34. In Fig. 103, AB = BC = CA = 2 cms., and the circular arcs touch the sides of AABC ; find the area of the figure. FIG. 103. 35*. A hoop, of radius 2', rests in a vertical position on a horizontal plane, with its rim in contact at A with a thin vertical peg, V high. The hoop is rolled over the peg into the corre- sponding position on the other side : Fig. 104 shows the area thus swept out. >f Calculate this area. FIG. 104. 36*. A triangular piece of cardboard ABC is such that BA = 8*, AC = 6*, /.BAG = 90. It is placed on the floor with the edge BC against the wall and a pin is put through the mid- point of BC. The cardboard is now turned about C till CA is against the wall, then about A till AB is against the wall, then about B till BC is against the wall; the cardboard remains in contact with the floor throughout. Construct the curve which the pin scratches on the floor and find the area between this curve and the wall. 37*. The diagonals AC, BD of the quadrilateral ABCD cut at right angles atO; AC -6", OC-OD-2*, OB = 4". The triangle DOC is cut away and the triangles AOD, BOC are 92 CONCISE GEOMETRY folded through 90 about OA, OB so as to form two faces of a tetrahedron on AOAB as base. Find (i) the volume of the tetrahedron ; (ii) the area of the remaining face ; (iii) the length of the perpendicular from O to the opposite face. 38*. ABCD is a rectangle; AB --= 10", AD = 6" ; AXB, BYC, CZD, DWA'are isosceles triangles, all the equal sides of which are 9" ; they are folded so as to form a pyramid with ABCD as base and X, Y, Z, W at the vertex. Find (i) the height of the pyramid ; (ii) the volume of the pyramid ; (iii) the total area of the surface of the pyramid. If AB=y, AD = /, AX ^r", and if the height^ of the pyramid = A*, prove that A 2 = r 2 Jp 2 - \q 2 . LOCI 93 LOCI THEOREM 45 A, B are two fixed points; if a variable point P moves so that PA = PB, then the locus of (or path trace^ out, by) P is the perpendicular bisector of AB. THEOREM 46 AOB, COD are two fixed intersecting lines ; if a variable point P moves so that its perpendicular distances PH, PK from these lines are equal, then the locus of (or path traced out by) P is the pair of lines which bisect the angles between AOB and COD. FIG. 106. DEFINITION. Given a point P and a line AB, if the per- pendicular PX from P to AB is produced to P 1 so that PX = XP 1 , then P 1 is called the image or reflection of P in AB, 94 CONCISE GEOMETRY LOCI EXERCISE XIX 1. A variable point is at a given distance from a given line, what is its locus 1 2. A variable point is at a given distance from a given point, what is its locus ? 3. A variable circle touches a fixed line at a fixed point, what is the locus of its centre 1 4. A variable circle passes through two fixed points, what i tue locus of its centre ? 5. A variable circle touches two fixed lines, what is the locus of its centre ? 6. A variable circle of given radius passes through a fixed point, what is the locus of its centre ? 7. A variable circle of given radius touches a fixed circle, what is the locus of its centre ? 8. A variable circle touches two fixed concentric circles, what is the locus of its centre ? 9. A variable circle of given radius touches a given line, what is the locus of its centre ? 10. PQR is a variable triangle; /.QPR*=90, PQ and PR pass through fixed points ; what is the locus of P ? 11. A, B are fixed points ; APB is a triangle of given area ; what is the locus of P ? 12. Given the base and vertical angle of a triangle, find the locus of its vertex. 13. A variable chord of a fixed circle is of given length, what is the locus of its mid-point ? 14. A is a fixed point on a fixed circle ; AP is a variable chord ; find the locus of the mid-point of A P. 15. P is a variable point on a given line ; O is a fixed point out- side the line ; find the locus of the mid-point of OP. 16. A, B are fixed points ; PAQB is a variable parallelogram of given area ; find the locus of P. 17. ABC is a given triangle ; BAPQ, CBQR are variable parallelo- LOCI 95 grams ; if P moves on a fixed circle, centre A, find the locus of R. 18. A variable chord PQ of a given circle passes through a fixed point ; find the locus of the mid-point of PQ. 19. The extremities of a line of given length move along two iixod perpendicular lines ; find the lorus of its mid point. 20. A, B are fixed points ; ABPQ is a variable parallelogram ; if AP is of given length, find the locus of Q. 21. PQ, QR are variable arcs of given lengths of a fixed circle, centre O ; PQ meets OR at S ; find the locus of S. 22. O, A are fixed points ; P is a variable point on OA ; OPQ is a triangle such that OP -f PQ is constant and /. OPQ is constant ; prove that the locus of Q is a straight line. 23. PQR is a variable triangle ; the mid-points of PQ and PR are fixed and QR passes through a fixed point; find the locus of P. 24. A, B are fixed points ; P moves along the perpendicular bisector of AB; AP is produced to Q so that AP= PQ ; find the locus of Q. 25. A, B are fixed points ; P is a variable point such that AP 2 4- PB 2 is constant ; find the locus of P. 26. A, B are fixed points ; P is a variable point such that PA 2 - PB 2 is constant ; prove that the locus of P is a straight line perpendicular to AB. 27. AB, AC arc two fixed lines; P is a variable point inside Z. BAC such that the sum of its distances from AB and AC is constant ; prove that the locus of P is a straight line. 28. A, B, C, D are fixed points ; P is a variable point such that the sum of the areas^f the triangles PAB, PCD is constant ; prove that the locus of P is a straight line. 29. If P 1 is the image of P in the line AB, prove that AP = AP X . 30. A variable line OQ passes through a fixed point O ; A is another fixed point; find the locus of the image of A in OQ. 31. A, B are two points on the same side of a line CD ; A 1 is the image of A in CD ; A*B cut CD at O ; prove that (i) AO and OB make equal angles with CD ; (ii) if P is any other point on CD, AP + PB > AO -f OB. 96 CONCISE GEOMETRY 32. AH, BK are the perpendiculars from A, B to XY. AH = BK = 7", HK = 16" ; what is the least value of AP + PB 1 B " H P K FIG. 107. 33. A, B are fixed points on opposite sides of a fixed line CD ; find the point P on CD for which PA~~PB has its greatest value. 34. How many images are formed when a candle is placed between two plane mirrors inclined to each other at an angle of (i)90; (ii)60? 35. If a billiard ball at A moves so as to hit a perfectly elastic cushion XY at P, it will continue in the line A*PB where A 1 is the image of A in XY; or, in other words, the two portions of its path AP and PB make equal angles with XY. ABCD is a rectangular billiard table with perfectly elastic cushions : a ball is at any point P ; it is struck in a direction parallel to AC ; prove that after hitting all four cushions it will again pass through P. THE TRIANGLEOONOURBENOY PROPERTIES 97 THR TRIANGLECONCURRENCY PROPERTIES THEOREM 47 If ABC is a triangle, the perpendicular bisectors of BC, CA, AB meet at a point O (say). A 3 X C Fio. 108. O is the centre of the circumcircle of the triangle ABC, and is called the circumcentre. THEOREM 48 If A .BO is a triangle, the internal bisectors of the angles ABC, BCA, CAB meet at a point I (say). B PC FIG. 109. I is the centre of the circle inscribed in the triangle ABC (i.e. the in-circle of AABC), and is called the in-centre. The external bisectors of the angles ABC, ACB meet at a point Ip which is the centre of the circle which touches AB produced, AC produced, BC; this circle is aaid to be escribed to BC, and \ l is called an ex-centre. 7 98 CONCISE GEOMETRY THEOREM 49 If ABC is a triangle, the altitudes AD, BE, CF meet at a point H (say). A H is called the orthocentre of the triangle ABC. The triangle DEF is called the pedal triangle of AABC. THEOREM 50 If ABC is a triangle, the medians AD, BE, CF meet at a point G (say), and DG- |DA. A BOO FIG. 111. Q is called the centro^l of the triangle ABC. THE TRIANGLE CONCURRENCY PROPERTIES 99 THE TRIANGLE CONCURRENCY PROPERTIES EXERCISE XX THE CIRCUMCIRCLK 1. If O is the circumcentre of A ABC and if D is the mid-point of BC, prove Z. BOD = Z. BAG. 2. The diagonals of the quadrilateral ABCD intersect at O; P, Q, R, S are the circumcentres of AS AOB, BOC, COD, DOA ; prove PQ = RS. 3. In AABC, /.BAC=90; P is the centre of the square described on BC ; prove that AP bisects /_ BAG. 4. In AABC, L BAG = 90; prove that the perpendicular bisectors of AB and AC meet on BC. 5. ABC is a scalene triangle; prove that the perpendicular bisector of BC and the bisector of /. BAG meet outside the triangle ABC. 6. ABCD is a parallelogram; E, F are the circumcentres of AS ABD, BCD ; prove that EBFD is a rhombus. 7. The extremities of a variable line PQ of given length lie on two fixed lines OA, OB ; prove that the locus of the circum- centre of AOPQ is a circle, centre O. 8. If the area of the triangle ABC is A, the radius of the circum- circle is ^-^ ; prove this for the case where /. B AC = 90. 9. ABCD is a quadrilateral such that AB = CD ; find a point O such that AOAB s AOCD. 10. AD, BE are altitudes of AABC; prove that the perpendicular bisectors of AD, BE, DE are concurrent. 11. In AABC, AB = AC; P is any point on BC; E, F are the circumcentres of AS ABP, ACP ; prove that AE is parallel to PF. 100 CONCISE GEOMETRY THE IN-CTRCLE AND EX-CIRCLES = a, CA-&, AB = c, and s = 12. In Fig. 112, if BC prove that (i) AY = s~a (ii) AQ^. (iii) BP-XC. (iv) YQ-ZR. (v) XP-6-c. (vi) IX = where A = area of triangle ABC. 3 (vii) , lP = * a . (viii) B, I, C, ^ are concyclic. (ix) AZ-f BX-f CY = ,s. (x) if L BIG -100, calculate FIG. 112. 13. AB is a chord of a circle; the tangents at A, B meet at T; prove that the in-centre of ATAB lies on the circle. 14. I is the in-centre and O the circumcentre of A ABC; prove that ZIAO = (ABC~Z.ACB). 15. I is the in-centre of A ABC ; prove that L AIC = 90 -f \ L ABC. 16. I is the in-centre and AD is an altitude of A ABC; prove that L IAD \( L ABC - L ACB). 17. In Fig. 112, prove that AB - AC = BX - XC. THE TRIANGLE CONCURRENCY PROPERTIES 101 18. The in-circle of A ABC touches BC at X, prove that the in-circles of AS ABX, ACX touch each other. 19. ABCD is a quadrilateral circumscribing a circle; prove that the in-circles of A ABC, CD A touch each other. 20. Two concentric circles are such that a triaiglo can be inscril>e<l in one and circumscribed to the other; piove that the triangle is equilateral. 21. In A ABC, Z.BAC = 90; prove that the diameter of the in-circle of AABC equals AB + AC - BC. 22. The extremities P, Q of a variable line lie on two fixed lines AB, CD; the bisectors of Z_s APQ, CQP meet at R; find the locus of R. 23. I is the in-centre of AABC; l x is the centre of the circle escribed to BC ; I, Ij cuts the circumcircle of AABC at P ; prove that I, \ lt B, C lie on a circle, centre P. 24. I is the in-centre of AABC; if the circumcircle of ABIC cuts AB at Q, prove AQ = AC. 25. I is the in-centre of AABC ; AP, AQ are the perpendiculars from A to Bl, Cl ; prove that PQ is parallel to BC. 26*. The in-circle of AABC touches BC, CA at X, Y; I is the in-centre; XY meets Al at P; prove Z.BPI = 90. THE ORTHOCENTRE 27. If AD, BE, CF are the altitudes of AABC and if H is its orthocentre (see Fig. 110), prove that (i) /.BHF^BAC. (ii) Z.BHC+Z.BAC = 180. (iii) AS AEF, ABC are equiangular, (iv) AS BDF, EDC are equiangular, (v) AD bisects L FDE. (vi) Z.EDF = 180-2 2 iBAC. (vii) H is in-centre of ADEF. 28. Where is the orthocentre of a right-angled triangle 1 29. Q is a point inside the parallelogram ABCD such that Z.QBC = 90= Z.QDC ; prove that AQ is perpendicular to BD. 30. If D is the orthocentre of AABC, prove that A is the ortho- centre of ABCD. 102 CONCISE GEOMETRY 31. If H is the orthocentre of A ABC, prove that the circum- circles of As AHB, AHC are equal. 32. I is the in-centre and l lf I 2 , I 3 are the ex-centres of AABC, prove that \ l is the orthocentre of Al I 2 ^ 33. In AABC, AB = AC, L BAG = 45; H is the orthocentre a: CHF is an altitude; prove that BF = FH. 34. O is the circumcentre and H the orthocentre of AABC ; prove that /. HBA = Z.OBC. 35. P, Q, R are the mid-points of BC, CA, AB ; prove that the orthocentre of APQR is the circumcentre of AABC. 36. H is the orthocentre of AABC ; AH meets BC at D and the circumcircle of AABC at P; prove that HD = DP. 37. O is the circumcentre, I is the in-centre, H is the orthocentre of AABC; prove that Al bisects Z.OAH. 38. BE, CF are altitudes of AABC; O is its circumcentre; prove that OA is perpendicular to EF. 39. H is the orthocentre and O the circumcentre of AABC ; AK is a diameter of the circumcircle ; prove that (i) BHCK is a parallelogram, (ii) CH equals twice the distance of O from AB. 40*. H is the orthocentre and O the circumcentre of AABC ; if AC = AH, prove /.BAG = 60. 41. H is the orthocentre of AABC ; BH meets the circumcircle at K; prove AH = AK. 42*. The altitudes BE, CF of AABC meet at H ; P, X are the mid-points of AH, BC; prove that PX is perpendicular to EF. 43. Given the base and vertical angle of a triangle, find the locus of its orthocentre. 44. [Nine Point Circle.] AD, B, CF are altitudes of AABC ; H is its orthocentre ; X, Y, Z, P, Q, R are the mid-points of BC, CA, AB, HA, HB, HC ; prove that (i) PZ is parallel to BE and ZX is parallel to AC. (ii) L PZX = 90 and L PYX - 90. (iii) P, Z, X, D, Y lie on a circle. (iv) The circle through X, Y, Z passes through P, Q, R, D, E, F. THE TRIANGLE CONCURRENCY PROPERTIES 103 THE CENTflOID 45. X, Y, Z are the mid-points of BC, CA, AB ; prove that the triangles ABC, XYZ have the same centroid. 46. ABCD is a parallelogram; P is the mid -point of AB; CP cuts BD at Q ; prove that AQ bisects BC. 47. If the medians AX, BY of AABC meet at G, prove that AS BGX, CGY are equal in area. 48. If G is the centroid of AABC and if AG = BC, prove that L BGC = 90. 49. If two medians of a triangle are equal, prove that the triangle is isosceles. 50. X, Y, Z are the mid-points of BC, CA, AB ; AD is an altitude of AABC ; prove that Z. ZXY = L ZD Y = Z. BAC. 51. AX, BY, CZ are the medians of AABC, prove that BY+CZ >AX. 52. If the centroid and circumcentre of a triangle coincide, prove that the triangle is equilateral. 53. ABCD is a parallelogram; H, K are the mid-points of AB, AD ; prove that CH and CK trisect BD. 54*. In a tetrahedron ABCD, the plane angles at each of three corners add up to 180 ; prove, by drawing the net of the tetrahedron, that its opposite edges are equal. RIDERS ON U0< > IV PROPORTION THKOBEM 51 If the heights AP, XQ of the triangles ABC, XYZ are equal, AABC^BC AXYZ YZ' A P C Y FKI. 113. THKOKKM 52 (1) If a straight line, drawn parallel to the base BC of the triangle ABC, cuts the sides AB, AC (produced if necessary) at H, K, ,, AH AK ,AH AK then - = - and = . HB KC AB AC (2) If H, K are joints on the sides AB, AC (or the sides produced) of the triangle ABC such that to BC. AH - A ~, HB KC , then HK is parallel H K B C Fia. H K FIG. 114(2). B C FIG. 114(3). 106 CONCISE GEOMETRY PROPORTION EXERCISE XXI 1. What is the value of the following ratios : (i) 3 ins. : 2 f t. ; (ii) U. : '2s. ; (iii) 20 min. : 1 hr. ; (iv) 3 sq. ft. : 2 sq. yd. ; (v) 3 right angles : 120 ; (vi) 3 m. : 25 cms. 1 2. Find x in the following : (i) 3 : ^ = 4 : 10, (ii) x feet : 5 yards = 2 . 3 ; (iii) 6 :x = x : 24 ; (iv) 2 hoars : 50 minutes = 3 shillings : x shillings. 3. If - = -, prove that b d /. v b d ,.. x , , /...v a, + b c-\-d (,)_=-; (u)ad-6e, (m) = _; 4. If = c = -, fill up the blank spaces in the following : b d f ,.^ a, c /..\ a b c d ,..., a + c . / T \ ( ' ~b + d+f v '6-3. -2 5,r 5. Solve the equations (i) "*-- = - ; (ii) & *~ 2 *' 6. Are the following in proportion (i) 3J, 5, 8, 12 ; (ii) 8 inches, 6 degrees, 1 2 degrees, 9 inches ? 7. Find the fourth proportional to (i) 2, 3, 4 ; (ii) at, be, cd. 8 Find the third proportional to (i) , \ ; (ii) J?, .ry. 9. Find a mean proportional between (i) 4, 25 ; (ii) a 2 6, be 2 . 10. A line AB, 8" long, is divided internally at P in the ratio 2:3; find AP. 11. A line AB, 8" long, is divided externally at Q in the ratio 7:3; find BQ. 12. AB is divided internally at C in the ratio 5 : 6. Is C nearer to A or B ! 13. AB is divided externally at D in the ratio 9:7. Is D nearer to A or B ? PROPORTION 107 14. AB is divided externally at D in the ratio 3 . 5. Is D nearer to A or B 1 15. A line AB, 6" long, is divided internally at P in the ratio 2:1, and externally at Q in the ratio 5.2; find the ratios in which PQ is divided by A and B. 16. ABODE is a straight line such that AB : BC : CD : DE-= 1.3.2-5 Find the ration (i) J| ; (ii) ^ , (iii) . Find the ratios in which BE is divided by A and D. If BE = 4", find AC. 17. A line AB, 8* long, is divided internally at C and externally at D in the ratio 7 . 3 ; O is the mid-point of AB ; prove that 00,00 -OB 2 . 18. A line AB, 6" long, is divided internally at C and externally at D in the ratio 4:1, O is the mid-point of CD , prove that AO = 1 6BO , and find the length of CD. 19. A line of length x" is divided internally in the ratio <i.l>\ find the lengths of the parts. 20. A line of length if is divided externally in the ratio a\b\ find the lengths of the parts. 21. A line AB is bisected at O and divided at P in tho ratio ,y;:y; OP find the ratio -. 22. AB is divided internally at C and externally at D in the ratio jc : y ; find (i) CD , (ii) the ratio in which B divides CD. 23. ABCDEF is a straight line such that AB : BC : CD : DE : EF =: p:r/:r:s:t', find (i) *?, (ii) ^, (m) the ratios in which A and E divide OF. If BD -a;", find AE. 24. ABCD, AXYZ are two straight lines such that AB : BC . CD = AX : XY : YZ. Fill up the blank spaces m the following : ,.. AB AC ,. a BC . , ; ..v XZ_ WAX"-"' (U) AD^AZ' (I11) AY~AC AR 25. ABC is a straight line; if AC = A. AB, find in terms of A. 26. The sides of a triangle are in the ratio x : y : z and its perimeter is p inches ; find the sides. 108 CONCISE GEOMETRY 27. AB is parallel to CD ; OB = 2", OD - 2 J", BC = 5* ; find AD. % a C D FIG. 115. 28. AB, CD, EF arc parallel lines; AC =-2', CE = 3", BF-4"; lind BD. A B FIG. 116. 29* - = ; AP, BQ, GN are perpendicular to OX ; OP -a, GB JJL OQ-6; find ON. Y Q / T _L X > P N Q FIG. 117. 30*. The medians of AABC intersect at G ; AP, BQ, CR, GN are the perpendiculars from A, B, C, G to a line OX ; if OP = a, OQ = b, OR = e ; prove ON = J(a 4- b + c). 31. ABC is a A; P, Q are points on AB, AC such that AP = JAB and CQ = JCA; prove that a line through C parallel to PQ bisects AB. 32. Three parallel lines AX, BY, CZ cut two lines ABC, XYZ ; prove that = PKOPOFTION 33. The diagonals of the quad. ABCD iuterse parallel to DC, prove = - . I- , O if AB is -rove f 1 a,t -a -. tj AB u 6 3 to cut 34. A line parallel ^o BC cuts AB, AC a AH.AC-AK.AB. 35. O is any point inside the AABC; a 1 ae XV cuts OA, OB at X, Y; YZ is drawi *ur,ii OC at Z ; prove XZ is parallel to AC. 36. ABCO is a quadrilateral ; P is anj po ; nt on AB ; lines PX, PY are drawn parallel to AC, AU to cut BC. 3D at X, Y ; prove XY is parallel to CD. 37. D is the foot of the perpendicular from A to the bisector of /.ABC; a line from D parallel to BC cuts AC at X; prove AX = XC. 38. In Fig. 118, prove - O FIG. 118. 39. I is the in-centre of AABC; prove that AlBC : AICA : = BC : CA : AB. A AQD AC 40. In Fig. 118, prove - = -. A Rf^I^ Rfi 41*. In Fig. 119, AH = HB, AK = 2KC; find the ratio of the areas of the small triangles in the figure; hence find the ,. CO ratio _. FIG. 119. 110 CONCISE GEOMETRY 42*. ABC is a A ; H, K are points on AB, AC such that HB = JAB and KC = JAC; BK cuts CH at O; prove BO = OK and CO = 2OH. [Use method of ex. 41.] 43*. ABC is a A ; Y, Z are points on AC, AB such that CY = $CA and AZ = JZB; BY cuts CZ at O; prove OY = |BY and OZ = fCZ. [Use method of ex. 41. J 44. Two circles APQ, AXY touch at A ; APX, AQY are straight r AP AQ lines; prove _ = ^. 45. ABCD is a parallelogram; any line through C cuts AB produced, AD produced at P, Q ; prove = ^ . BP DA 46*. ABCD is a parallelogram ; a line through C cuts AB, AD, BD (produced if necessary) at P, Q, O ; prove OP . OQ = OC 2 . 47. ABC is a A; three parallel lines AP, BQ, CR meet BC, CA, AB (produced if necessary) at P, Q, R ; prove that B _ P x99 x *?=l PC QA RB 48*. O is any point inside AABC ; D, E, F are points on BC, CA, AB such that AD = BE = CF; lines are drawn from O parallel to AD, BE, CF to meet BC, CA, AB at P, Q, R ; prove OP -f OQ + OR = AD. 49*. ABC is a triangle; a line cuts BC produced, CA, AB at P, Q, R ; CX is drawn parallel to PQ, meeting AB at X ; ... BP BR .... BP CQ AR , prove (i) pc = R- x ; (n) pc x QA x RB ==1 - [This is known as Menelaus' Theorem.] SIMILAR TRIANGLES 111 SIMILAR TRIANGLES THEOREM 53 If the triangles ABC, XYZ are equiangular (/.ABC= Z.XYZ and Z.ACB-ZXZY), ,, AB BC CA then = = . c. Fro. 120. THEOREM 54 AR BC CA If the triangles ABC, XYZ are such that ^= ~ -^-, then the X I I fm 21 /V triangles are equiangular, Z. ABC = Z. XYZ, L ACB = Z. XZY, ZlBAC= Z.YXZ. THEOREM 55 If, in the triangles ABC, XYZ, Z.BAC-Z.YXZ and ^ = ^? XY XZ> then the triangles are equiangular, Z.ABC= /.XYZ and Z.ACB=ZXZY. 112 CONCISE GEOMETRF SIMILAR TRIANGLES EXERCISE XXII 1. A pole 10' high casts a shadow 3J' long; at the same time a church spire casts a shadow 42' long. What is its height ? 2. In a photograph of a chest of drawers, the height measures 6" and the breadth 3*2* ; if its height is 7J feet, what is its breadth ? 3. Show that the triangle whose sides are 5'1", 6*8", 8*5" is right-angled. 4. A halfpenny (diameter 1") at the distance of 3 yards appears nearly the same size as the sun or moon at its mean dis- tance. Taking the distance of the sun as 93 million miles, find its diameter. Taking the diameter of the moon as 2160 miles, find its mean distance. 5. How far in front of a pinhole camera must a man 6' high stand in order that a full-length photograph may be taken on a film 2 " high, 2 J* from the pinhole ? 6. The slope of a railway is marked as 1 in 60. What height (in feet) does it climb in f mile 1 7. A light is 9' above the floor ; a ruler, 8* long, is held horizon- tally 4' above the floor ; find the length of its shadow. 8. Two triangles are equiangular \ the sides of one are 5", 8", 9" ; the shortest side of the other is 4 cms. ; find its other sides. 9. The bases of two equiangular triangles are 4*, 6* ; tho height ef the first is 5* ; find the area of the second. 10. In AABC, AB = 8", BC = 6*, CA = 5" ; a line XY parallel to BC cuts AB, AC at X, Y ; AX = 2" ; find XY, CY. 11. In quadrilateral ABCD, AB is parallel to DC and AB = 8", AD = 3", DC = 5" ; AD, BC are produced to meet at P ; find PD. 12. A line parallel to BC meets AB, AC at X, Y; BC = 8*, XY = 5" ; the lines BC, XY are 2'" apart. Find the area of A AXY. SIMILAR TRIANGLES 113 13. In Fig. 121, (i) if AO = 3*, OB = 2", AB = V, DC 1 J", find CO, DO, (ii) if AO^S", BO = 4*, AC^T", find BD. (iii) if PA = 9", PB^S", AB = <r, PC = 3* find PD, CD. (iv) if PA = 9", PB = 8", AC = 6", PC = 4*, find BD, D. P 14. Show that the line joining (1, 1) to (4, 2) is parallel to and half of the line joining (0, 0) to (6, 2). 15. Three lines APB, AQC, ARD are cut by two parallel lines PQR, BCD ; AR = 3*, RD = 2", BC = 4" ; find PQ. 16. In Fig. 122, AB is parallel to OD ; AB = 6', BO = 20', BE = 5', DQ = 9 ; ; findOD, BP. A Q B D FIG. 122. 17. The diameter of the base of a cone is 9" and its height is 15* ; find the diameter of a section parallel to the base and 3" from it. 18. AXB is a straight line ; AC, XY, BD are the perpendiculars from A, X, B to a line CD; AC = 10, BD = 16, AX = 12, XB = 6; findXY. 19. A, B are points on the same side of a line OX and at distances 1", 5* from it ; Q and R divide AB internally and externally in the ratio 5:3; find the distances of Q and R from OX. 8 114 CONCISE GEOMETRY 20. A rectangular table, 5' wide, 8' long, 3' high, stands on a level floor under a hanging lamp ; the shadow on the floor of the shorter side is 8' long; find the length of shadow of the longer side and the height of the lamp above the table. 21. A sphere of 5" radius is placed inside a conical funnel whose slant side is 12" and whose greatest diameter is 14"; find the distance of the vertex from the centre of the sphere. 22. The length of each arm of a pair of nutcrackers is 6" ; find the distance between the ends of the arms when a nut 1" in diameter is placed with its nearer end 1" from the apex. 23. In Fig. 123, PQBR is a rectangle. (i) If AB = 7, PQ - 1, PR = 2, find BC. (ii) If AB = 7, BC = 5, PR = #, PQ=y, find an equation between #, y. A K B Q C FIG. 123. 24. In AABC, Z. ABC = 90, AB - 5", BC = 2" ; the perpendicular bisector of AC cuts AB at Q ; find AQ. 25. The diameter of the base of a cone is 8* ; the diameter of a parallel section, V from the base, is 6" ; find the height of the cone. 26. In Fig. 124, AB, PN, DC are parallel; AB = 4*, BC = 5*, CD = 3" ; calculate PN. FIG. 124. SIMILAR TRIANGLES 115 27. ABCD is a quadrilateral such that L ABC = 90 = L AGO, AC -5", BC = 3", 00 = 10"; calculate the distances of D from BC, BA. 28. PQ is a chord of a circle of length 5 cms.; the tangents at P, Q meet at T ; PR is a chord parallel to TQ ; if PT = 8 eras., find PR. 29. (i) A - man, standing in a room opposite to arid 6' from a window 27" wide, sees a wall parallel to the plane of the window. With one eye shut, he can see 18" less length of wall than with both eyes open; supposing his eyes are 2" apart, find the distance of the wall from the window and the total length of wall visible. (ii) If the window is covered by a shutter containing a vertical slit \" wide, show that there is a part of the wall out of view which lies between two parts in view and find its length. (iii) A man in bed at night sees a star pass slowly across a vertical slit in the blind ; shortly afterwards, this occurs again, la it possible that he sees the same star twice 1 Explain your answer by a figure. 30. A rectangular sheet of paper ABCD is folded so that D falls on B; the crease cuts AB at Q ; AB = 11", AD = 7"; find AQ. 31. Fig. 125 represents an object HK and its image PQ in a con- cave mirror, centre O, focus F. CH-w, CP-v, CF = FO=/ ., , ,. v 1 1 1 ,.. x vx prove that (i) - = -- ; (u) y = . 116 CONCISE GEOMETRY 32. In Fig. 126, with the same notation as in ex. 31, prove that ~ = - 4- -, and find y in terms of x, u, f. t u v 1. K ""-- I O P r~^ H;: ^i FIG. 126. 33. Fig. 127 represents an object HK and its image PQ iu a thin concave lens, centre O, focus F. prove that (i) -- = ?--- ; (ii) y - . / V U U r-7 K B\ -A* - z- : : : ; Q- : : - - -i 6 P "H *F FIG. 127. 34. Fig. 128 represents an object HK and its image PQ in a thin convex lens, centre O, focus F. OH = u. OP = v. OF =f, HK = x. PO = v ' j ^" > ** m j t "* ^i \e ii j prove that - = - + -, and find y in terms of #. u, f. f u v .13 J< ...P....r::ll\Li-----' - "*""""'"" I ]V:;-----'--~T r ]o "" H" cf" v FIG.. 128. 35. OAOB is a quadrilateral on level ground; Z.AOB = 90 = Z.OBC, ZLOAC=135, OB = 9', OA = 12'; it is covered by a plane roof resting on pillars ; the pillars at A, B are 10' high, the pillar at O is 8' high ; find the height of the pillar atC. SIMILAR TRIANGLES 117 36. AB, DC are the parallel sides of a trapezium ABCD ; the diagonals cut at O ; prove = -- OC CD 37. BE, CF are altitudes of AABC; prove - = . CF AC 38. AOB, COD are two intersecting chords of a circle ; fill up the blank spaces in (i) * = ^ ; (ii) ~ - - . 39. Two straight lines OAB, OCD cut a circle at A, B, C, D ; fill up the blank spaces in (i) = ; (ii) - = . BD OC 40. ABC is a A inscribed in a circle ; the bisector of L BAG cuts BC at Q and the circle at P ; prove = ^ and complete l AP AB BO PC the equation -= = ~ . 1 AB 41. In AABC, Z.BAC = 90; AD is an altitude; prove that DC AC CD __. ~ and complete the equation _ = -. AC BC F H DA DB 42. The medians BY, CZ of AABC intersect at G; prove that 43. BE, CF are altitudes of AABC; prove that ~ =^. 44. Two lines AOB, POQ intersect at O ; the circles AOP, BOQ cut again at X ; prove that = - . ,/vp '^y 45. Prove that the common tangents of two non-intersecting circles divide (internally and externally) the line joining the centres in the ratio of the radii. 46. M is the mid-point of AB ; AXB, MYB are equilateral tri- angles on opposite sides of AB ; XY cuts AB at Z ; prove 47. AB is a diameter of a circle ABP ; PT is the perpendicular PT AP from P to the tangent at A ; prove = . 48. APB, AQB are two circles ; if PAQ is a straight line, prove BP that equals the ratio of their diameters. BQ 118 CONCISE GEOMETBY 49. ABCD is a parallelogram ; any Hue through C cuts AB produced, AD produced at X, Y ; prove = . BX AB 50. ABCD is a rectangle ; two perpendicular lines are drawn ; one cuts AB, CD at E, F ; the other cuts AD, BC at G, H ; EF BC prove - .= . 1 GH AB 51. In the quadrilateral ABCD, L ABC -/ ADC and ^?^ CD ; BC DA prove AB = CD, 52. The diagonals AC, BD of the quadrilateral ABCD meet at O ; if the radius of the circle AOD is three times the radius of the circle BOC, prove AD^3BC. 53. ABCD is a parallelogram ; P is any point on AB ; DP cuts AC at O ; prove - = -. V ' * AB DQ 54. AB, DC are the parallel sides of the trapezium ABCD ; any line parallel to AB cuts CA, CB at H, K ; DH, DK cut AB at X, Y ; prove AB - XY. 55. ABCD is a parallelogram ; O is any point on AC ; lines POQ, ROS are drawn, cutting AB, CD, BC, AD at P, Q, R, S ; prove PS is parallel to QR, 56. In A ABC, D is the mid-point of BC ; AD is bisected at E; BE cuts AC at F; prove CF= 2FA. [Draw EK parallel to BC to cut AC at K.] 57. BC, YZ are the bases of two similar triangles ABC, XYZ ; AP, XQ are medians ; prove L BAP = L YXQ. 58. P is a variable point on a given circle ; O is a fixed point outside the circle \ Q is a point on OP such that OQ = $OP ; prove that the locus of Q is a circle. 59. ABC is a A ; E, F are the mid-points of AB, AC ; EFD is drawn so that FD -= 2EF ; prove BF bisects AD. 60. In AABC, Z BAC = 90; ABXY, ACZW are squares outside AABC ; BZ, CX cut AC, AB at K, H ; prove AH - AK. 61. In AABC, the bisectors of /.a ABC, ACB meet at D; DE, DF are drawn parallel to AB^ AC to meet BC at E, F ; prove BE_BA FC~AC* SIMILAR TRIANGLES 119 62*. Tn AABC, Z. BAC = 90 ; AD is an altitude ; H, K are the in-centres of AS ADB, ADC ; prove that AS DHK, ABC are similar. 63*. D, E, F are the mid-points of the sides BC, CA, AB of a triangle ; O is any other point ; prove that the lines through D, E, F parallel to OA, OB, OC are concurrent. 64*. In AABC, ABa . AC ; BQ is the perpendicular from B to the bisector of / BAC ; BC cuts AQ at P ; prove? that PQ ^ - 1 PA " 2 120 CONCISE GEOMETRY RECTANGLE PEOPERTIES OF A CIRCLE THEOREM 56 (i) If two chords AB and CD of a circle intersect at a point O (inside or outside a circle), then OA.OB = OC.OD. ,0 FIG. 129(1). FIG. 129(2). (ii) If from any point O outside a circle, a line is drawn touching the circle at T, and another line is drawn cutting the circle at A, B, then OA.OB = OT 2 . O FIG. 130. RECTANGLE PROPERTIES OF A CIRCLE 121 THEOKEM 57 If AD is an altitude of the triangle ABC, which is right-angled at A, then (i) AD 2 = BD . DC ; (ii) BA 2 = BD . BC. FIG. 131. DEFINITION. If a, #, b are such that ~ = ~ or x* = ab, x b x is called the mean proportional between a and b. The converse properties are important : (i) If two lines AOB, COD are such that AC . OB = CO . OD, then A,B, C, D lie on a circle. (ii) If two lines OAB, ODC are such that OA . OB = OC . OD, then A, B, C, D lie on a circle. (iii) If two lines OBA, OT are such that OA . OB = OT 2 , then the circle through A, B, T touches OT at T. Alternative proof of Theorem 57 : (i) Draw the circle on BC as diameter : it passes through A, since L BAG = 90. Produce AD to cut the circle again at E. Since the chord AE is perp. to diameter BC, AD = DE. But AD.DE=BD.DC; AD 2 =BD.DC. (ii) Draw the circle on AC as diameter : it passes through D, since Z.ADC = 90, and touches BA at A, since /. by Theorem 56 (ii), BA 2 = BD . BC. 122 CONCISE GEOMETRY RECTANGLE PROPERTIES OF A CIRCLE EXERCISE XXIII 1. Find a mean proportional between (i) 3 and 48 ; (ii) I2x, 2. From a point P on a circle, PN is drawn perpendicular to a diameter AB ; AN - 3", NB -= 12" ; find PN. 3. In A ABC, Z BAG -00; AD is an altitude; AB = r>", AC = 12"; find BD. 4. In A ABC, AB-8, AC -12; a circle through B, C cuts AB, AC at P,Q ; BP = 5 ; find CQ. 5. The diagonals of a cyclic < quadrilateral ABCD meet at O; AC = 9, BD = 12, OA - 4 ; find OB. G. In Fig. 132, (i) If AB - 9, BO = 3, find OT. (ii) If OB - 6, OT- 12, find AB. (Hi) If OA= 3, AB - 2, AT- 4, find BT. (iv) If AB = 8, AT -6, BT-5, find OT. 7. ABC is a triangle inscribed in a circle; AB = AC = 10" BC = 12" ; AD is drawn perpendicular to BC and is produced to meet the circle in E ; find DE and the radius of the circle. 8. In A ABC, Z ABC = 90, AB = 3", BC = 4"; find the radius of the circle which passes through A and touches BC at C. 9. In A ABC, Z.BAC-9CT; AD is an altitude ; BC-a, CA = 6, AB = c, AD = A, BD = x, DC = y ; prove that (i) A 2 = xy ; (ii) i'=ry (* + y); (iii) he * b* ; (iv) X 10. In Fig. 132, if OA-2OT, prove AB-3BO. RECTANGLE PROPERTIES OF A CIRCLE 123 11. AOB, COD are two perpendicular chords of a circle, centre K; AO = 6, CO = 10, OD = 12 ; find OK, AK. 12. X is the mid-point of a line TY of length 2" ; TZ is drawn so that /. ZTX = 45 ; a circle is drawn through X, Y touching TZ at P ; prove /1TXP = 90, and find the radius of tlie circle. 1 3. ABC is a A inscribed in a circle ; the tangent at C meets AB produced in D; BC=^jp, CA-</, AB^r, BD = #, CD=-y; find x, y in terms of p, <?, r. 14. Express, in the form of equal ratios, the equations : (i) xy^ab\ (ii) pq = r 2 ; (iii) OA . OB = OC . OD ; (iv) ON . OT - OP 2 . 15. The diagonals of a cyclic quadrilateral ABCD intersect at O ; prove AD . OC = BC . OD. 16. Two lines OAB, OCD cut a circle at A, B, C, D ; prove OA . BC - OC . AD. 17. Two chords AB, CD of a circle intersect at O ; if D is the mid-point of arc AB, prove CA . CB = CO . CD. '18. In A ABC, AB - AC and L BAC = 36 ; the bisector of L ABC meets AC at P ; prove AC . CP = BC 2 - AP 2 . 19. The altitudes BE, CF of A ABC intersect at H ; prove that (i) BH.HE = CH.HF; (ii) AF . AB - AE . AC ; (iii) CE.CA = CH.CF. 20. In AABC, AB = AC ; D is a point on AC such that BD = BC ; prove BC 2 = AC . CD. 21. Two circles intersect at A, B ; P is any point on AB produced ; prove that the tangents from P to the circles are equal. 22. In AABC, Z.BAC = 90, AB = 2AC; AD is an altitude; prove BD = 4DC. 23. PQ is a chord of a circle, centre O ; the tangents at P, Q meet at T ; OT cuts PQ at N ; prove ON . OT = OP 2 . 24. AB is a diameter of a circle ; PQ is a chord ; the tangent at B meets AP, AQ at X, Y ; prove AP . AX = AQ . AY. 25. AB, AC are two chords of a circle ; any line parallel to the tangent at A cuts AB, AC at D, E ; prove AB . AD = AE . AC. 26. ABCD is a cyclic quadrilateral; P is a point on BD such that L PAD = L BAG ; prove that (i) BC . AD AC . DP ; (ii) AB.CD = AC.BP; (iii) BC . AD + AB . CD- AC . BD. 124 CONCISE GEOMETRY 27. AB is a diameter of a circle, centre O; AP, PQ are equal chords ; prove AP . PB = AQ . OP. 28. AD is an altitude of A ABC ; prove that the radius of the circle ABC equals "- . [Draw diameter through A.] 29. Two circles intersect at A, B ; PQ is their common tangent ; prove AB bisects PQ. 30. In AABC, AC is equal to the diagonal of the square de- scribed on AB ; D is the mid-point of AC ; prove Z. ABD = Z ACB. 31. A line PQ is divided at R so that PR 2 = PQ. RQ; TQR is a A such that TQ = TR = PR ; prove PT = PQ. 32. PQR is a A inscribed in a circle; the tangent at P meets TO PO 2 QR produced at T; prove =r _ T 33. In AABC, ZBAC = 90; E is a point on BC such that AE = AB ; prove BE . BC = 2AE 2 . 34. AD is an altitude of AABC; if AB.BC--=AC 2 and if AB = CD, prove Z.BAC = 90. 35. Two chords AB, AC of a circle are produced to P, Q so that AB = BP and AC = CQ ; if PQ cuts the circle at R, prove AR 2 = PR.RQ. 36 The tangent at a point C on a circle is parallel to a chord DE and cuts two other chords PD, PE at A, B; prove AC = AD CB "~ BE' 37. AB is a diameter of a circle, centre O ; the tangents at A, B meet any other tangent at H, K ; prove AH . BK = AO 2 . 38. Two lines OAB, OCD cut a circle at A, B, C, D ; through O, a line is drawn parallel to BC to meet DA produced at X ; prove XO 2 = XA.XD. 39. ABC is a A inscribed in a circle ; a line through B parallel to AC cuts the tangent at A in P ; a line through C p v f AP AB 2 1 to AB cuts AP in Q ; prove - = - . i v ' ^ AQ AC 2 j 40*. AB is a chord of a circle APB ; the tangents at A, u \ ( at T ; PH, PK, PX are the perpendiculars to TA, TB, AB ; prove PH. PK-PX 2 . RECTANGLE PKOPERTIES OF A CIRCLE 125 41*. AB, AC are tangents to the circle BDCE ; ADE is a straight line ; prove BE . CD = BD . CE. 42*. P, Q are points on the radius OA and OA produced of a circle, centre O, such that OP . OQ = OA 2 ; R is any other point on the circle ; prove that RA bisects Z. PRQ. 43*. In AABC, AB-AC, Z.BAC = 36; prove AB 2 -BC 2 = AB . BC. 44*. The internal bisector of /.BAG cuts BC ut D, prove that AD 2 = BA . AC - BD . DC. [Use ex. 1 7.] 45* The external bisector of Z.BAC cuts BC produced at E; prove that AE 2 = BE . EC - BA . AC. 46*. ABCD is a parallelogram ; H, K are fixed points on AB, AD ; HP, KQ are two variable parallel lines cutting CB, CD at P, Q ; prove BP, DQ is constant. 126 CONCISE GEOMETRY AREAS AND VOLUMES THKOEEM 58 If ABC, XYZ are two similar triangles, and if EC, YZ are a pair of corresponding sides, then AABC^BC* AXYZ YZ2 H C FKI. 133. K Z More generally, the ratio of the areas of any two similar polygons is equal to the ratio of the squares on corresponding sides. THEOREM 59 If AB and CD are corresponding sides of any two similar polygons PAB, QCD, and if AB, CD, EF are three lines in proportion ,. AB CD, (i.e. ~ ), V CD EF'' The following facts are also of importance (see ex. 34, 35) : (i) The ratio of the areas of the surfaces of similar solids equals the ratio of the squares of their linear dimensions, (ii) The ratio of the volumes of similar solids equals the ratio of the cubes of their linear dimensions. AREAS AND VOLU]\<ttS 127 AREAS AND EXEBCISIJ XXIV 1. A screen, 6' high (not necessarily reeUM, iPr i 'quires L ; 7 sq. ft. of material for covering : h-v mm '; s uf ;ded for a screen of the same shape, 4' liig} ? 2. On a map whose scale is 6" to the mile, a plot of ground is represented by a triangle of area 2 J ;;q. inches ; what is the area (in acres) of the plot 1 3. The sides of a triangle are 6 cms., 9 cms., 12 cms. ; how many triangles whose sides are 2 cms., 3 cms., 4 cms. can be cut out of it 1 How would you cut it up ? 4. Show how to divide any triangle into 25 triangles similar to it. 5. The area of the top of a table, 3 feet high, is 20 sq. ft. ; the area of its shadow on the floor is 45 sq. ft. ; find the height of the lamp above the floor. 6. A light is 12 feet above the ground; find the area of the shadow of the top of a table 4 ft. high, 9 ft. long, 5 ft. broad. 7. ABC, XYZ are similar triangles; AD, XK are altitudes; AB = 15, BC - 14, CA = 13, AD = 12, XY = 5 ; find XK and the ratio of the areas of AS ABC, XYZ. 8. A triangle ABC is divided by a line HK parallel to BC into two parts AHK, HKCB of areas 9 sq. cms., 16 sq. cms. ; BC-7 cms.; find HK. 9. E is the mid-point of the side AB of a square ABCD ; AC cuts ED at O ; AB = 3" ; find the area of EBCO. 10. ABC is a A such that AB = AC = 2BC; D is a point on AC such that Z. DBC = Z. BAC ; a line through D parallel to BC cuts AB in E ; find the ratio of the areas AABC : ABCD : ABED : AEDA. 11. Water in a supply pipe of diameter 1 ft. comes out through a tap $" in diameter : in the pipe it is moving at 5* a second ; with what velocity does it come out of the tap ? 12. If it costs 3 to gild a sphere of radius 3 ft., what will it cost to gild a sphere of radius 4 ft. 1 13. Two hot-water cans are the same shape ; the smaller is 9* high 128 CONCISE GEOMETRY and holds a quart ; the larger is 15* high : how much will it hold 1 14. How many times can a cylindrical tumbler V high and 3* in diameter be filled from a cylindrical cask 40* high and 30* in diameter 1 15. A metal sphere, radius 3*, weighs 8 Ib. ; find the weight of a sphere of the same metal 1' in radius. 16. A cylindrical tin 5* high holds | Ib. of tobacco ; how much will a tin of the same shape 8* high hold ? 17. Two models of the same statue are made of the same material ; one is 3* high and weighs 8 oz. ; the other weighs 4 Ib. ; what is its height 1 18. A lodger pays 8 pence for a scuttle of coal, the scuttle being 20" deep ; what would he pay if the scuttle was the same shape and 2 feet deep. 19. A tap can fill half of a spherical vessel, radius 1| feet, in 2 minutes; how long will two similar taps take to fill one- quarter of a spherical vessel of radius 4 feet ? 20. Two leaden cylinders of equal lengths and diameters 3*, 4 are melted and recast as a single cylinder of the same length what is its diameter ? 21. In the given figure, not drawn to scale, the lines AB, CD bisect each other at right angles ; AB = 6 cms., CD = 4 cms., PAQ, RBS are arcs of circles of radii 1 cm. ; PCS, QDR are arcs of circles of radii 3J cms., touching the former arcs. Construct a similar figure in which the length of the line corresponding to AB is 9 cms. The area of the first figure is approximately 18 sq. cms., what is the area of the enlarged figure ? ABEAS AND VOLUMES 129 If in the given figure, tho curve is rotated about AB to form an egg-shaped solid, its volume Is approximately 48 c.c. ; what is the volume of the solid obtained similarly from the enlarged figure ? 22. The sides of a AABC are trisected as in the figure ; prove that the area of PQRSXY =-- f A ABC. A /\ X .A B P Q C Fro. 136. 23. If in the AS ABC, XYZ, Z.BAC=: Z.YXZ, 4.1 4. AABC AB . AC prove that ^ = . F A XYZ XY.XZ 24. Two lines CAB, OCD meet a circle at A, B, C, D, prove that A PAD AD \yh a t result is obtained by making B co- AOBC BC 2 J 8 incide with A ? 25. H, K are any points on the sides AB, AC of AABC, prove 4.1 4. AAHK AH . AK that ^ = . AABC AB . AC 26. In AABC, Z.BAC = 90 and AD is an altitude ; AB* BD prove = -- . 1 AC 2 DC 27. ABCD is a parallelogram ; P, Q are the mid-points of CB, CD ; prove AAPQ = f parallelogram ABCD. 28. Any circles through B, C cuts AB, AC at D, E; prove AADE^DE 2 AABC~BC 2 ' 29. In AABC, Z.BAC = 90 and AD is an altitude; DE is the ,. i * ~ . * BE BA 2 perpendicular from D to AB ; prove = 5. 30. AP is a chord and AB is a diameter of a circle, centre O ; the tangents at A, P meet at T ; prove -= jj- 130 CONCISE GEOMETRY 31. ABC is an equilateral triangle; BC is produced each way to PR AP 2 P, Q; if ZPAQ- 120, prove ~ j^- 32. In AABC, Z BAG = 90; BCX, CAY, ABZ are similar triangles with X, Y, Z corresponding points; prove A CAY -f A ABZ -= A BCX. 33. A room is lighted by a single electric bulb in tflie ceiling; a table with level top is moved about in the room ; prove that the area of the shadow of the top on the floor does not alter. 34. If ,r ins. is the length of some definite dimension in a figure of given shape, its area^ta? 2 sq. ins. where Ic is constant for different sizes. Find k for (i) sqnare,iside #; (ii) square, diagonal x\ (iii) circle, radius x\ (iv) circle, perimeter x -, (v) equilateral triangle, side x\ (vi) regular hexagon, side x ; (vii) surface of cube, side x ; (viii) surface of sphere, radius jr. 35. If x ins. is the length of some definite dimension in a figure of given sha}>e, its volume = Avr 8 cu. ins. where k is constant for different sizes. Find /< for (i) cube, edge jc ; (ii) cube, diagonal x ; (iii) sphere, diameter x ; (iv) sphere, equator x ; (v) the greatest circular cylinder that can be cut from a cube, edge x ; (vi) circular cone, vertical angle 90, height x ; (vii) regular tetrahedron, edge x. BISECTOR OF VERTICAL ANGLE OF A TRIANGLE 131 THE BISECTOR OF THE VERTICAL ANGLE OF A TRIANGLE THEOREM 60 (i) ABC is a triangle ; if the line bisecting /. BAG (internally or externally) cuts BC, or BC produced at D, ,, BA BD then si- AC DC (ii) If D is a point on the base BC, or BC produced of the triangle ABC such that , then AD bisects internally AC DC or externally /. BAG. A DC B C FIG. 137(1). FIG. 137(2). 132 CONCISE GEOMETRY THE BISECTOR OF THE VERTICAL ANGLE OF A TRIANGLE EXERCISE XXV 1. In AABC, AB = 6 cms., BC = 5 cms., CA = 4 cms.; the internal ami external bisectors of Z.BAC cut BC and BC produced at P, Q; find BP and BQ and show that L+ JU1. BP BQ BC* 2. In AABC, AB = 4", BC = 3* CA-5"; the bisector of ZACB cuts AB at D ; find CD. 3. In AABC, AB=12, BC = 15, CA = 8; P is a point on BC such that BP = 9 ; prove AP bisects /. BAG ; if the external bisector of /. BAC cuts BC produced at Q, and if D is the mid-point of BC, prove that DP . DQ = DC 2 . 4. The internal and external bisectors of /. BAC meet BC and BC produced at P, Q ; BP = 5, PC = 3 ; find CQ. 5. ABCD is a rectangular sheet of paper ; AB = 4*, BC = 3" ; the edge BC is folded along BD and the corner is then cut off along the crease ; find the area of the remainder. 6. In AABC, AB = 6*, AC = 4"; the bisector of /.BAG meets the median BE at O; the area of AABC is 8 sq. in.; what is the area of A AOB ? 7. The internal and external bisectors of Z.BAC cut BC and oo RO BC produced at P, Q : prove = ~-?. F '*' F PC CQ 8. AX is a median of AABC; the bisectors of /.a AXB, AXC meet AB, AC at H, K ; prove HK is parallel to BC. 9. ABCD is a parallelogram ; the bisector of /. BAD meets BD at K; the bisector of Z.AB'C meets AC at L; prove LK is parallel to AB. BISECTOR OF VERTICAL ANGLE OF A TRIANGLE 133 10. The tangent at a point A of a circle, centre O, meets a radius OB at T ; D is the foot of the perpendicular from A to OB ; 11. The bisector of Z.BAC cuts BC at D ; circles with B, C as centres are drawn through D and [cut 8A, CA at H, K ; prove HK is parallel to BC. 12. H is any point inside the A ABC; the bisectors of /.s BHC, CHA, AHB cut BC, CA, AB at X, Y, Z; prove BX CY AZ , ___ N/ -- . \f __ = I XC YA ZB 13. Two lines OAB, OCD meet a circle at A, B, C, D ; the bisector of ZAOC cuts AC, BD at H, K; prove ^J = ?^. HC KB 14. The bisector of /.BAC cuts BC at D; the circle through A, B, D cuts AC at P ; the circle through A, C, D cuts AB at Q ; prove BQ = CP. 15. Two circles, centres A, B, touch at O; any line parallel to AB cuts the circles at P, Q respectively; AP and BQ are produced to meet at K ; prove OK bisects /. AKB, 16. A straight line cuts four lines OP, OQ, OR, OS at P, Q, R, S ; if ^POR-90 and OR bisects Z.QOS, prove ^= j 1 7. The tangent at a point T on a circle cuts a chord PQ when produced at O; the bisector of /.TOP meets TP, TQ at X, Y ; prove TX 2 = TY 2 = PX . QY. 18. In A ABC, Z. BAC = 90 and AD is an altitude; the bisector of /.ABC meets AD, AC at L, K; prove ^t = C -. LD KA 19. ABCD is a quadrilateral; if the bisectors of Z.s DAB, DCB meet on DB, prove that the bisectors of /.s ABC, ADC meet on AC. 20. Two circles touch internally at O ; a chord PQ of the larger QP PR touches the smaller at R : prove = - - . OQ RQ . If I is the in-centre of A ABC, and if Al meets BC at D, prove AIAB + AC 134 CONCISE GEOMETRY 22*. The internal and external bisectors of Z.APB meet AB at X, Y; prove ^XPY = 90. If A, B are fixed points and if P varies so that is constant, prove that the locus of PB P is a circle. \Apolloniu3 circle.] 23*. If the internal and external bisectors of /. BAG meet BC and BC produced at D, E, prove DE 2 = EB . EC - DB . DC. 24*. ABC is a triangle such that AB4-AC = 2BC ; the bisector of L BAG meets BC at D ; prove AD 2 = 3BD . DC. EXAMPLES ON THE CONSTRUCTIONS OF BOOK 1 USE OF INSTRUMENTS EXERCISE XXVI USE OF RULER, DIVIDERS, AND PROTRACTOR 1. Measure in inches and cms. the lines a, 6, c, d. FIG. 138(1). 2. Draw a straight line across your sheet of paper arid mark off by eye lengths of 4 cms., 7 cms., 2 inches; then measure them and write down your errors. 3. Draw a line and cut off from it a length of 5" ; measure it in cms. and find the number of cms. in 1 inch. 4. Draw a line and cut off from it a length of 10 cms. ; measure it in inches and hence express 1 cm. in inches. 5. In Fig. 138(2), measure in cms. the lengths of AC, BD, BC, AD. What are the values of (i) AC + BD ; (ii) AD + BC. B FIG. 138(2). 135 136 CONCISE GEOMETRY 6. Measure in inches and cms. the length of this page. Taking 1" = 2'54 cms. approx., find how far your measurements agree with each other. 7. Draw a straight line across your paper : mark the middle point by eye and measure the two parts. How far is the point you have marked from the real mid-point of the line 1 8. Draw a straight line across your paper and divide it by eye into three equal parts : measure the three parts. 9. Rei>eat ex. 8, dividing the line into four equal parts. 10. Draw a straight line across your paper and use your dividers (i) to bisect it ; (ii) to trisect it. 11. It is required to obtain points on a line AB produced beyond an obstacle which obstructs the view. C is one of the points required, perform the construction and verify it. FIG. 139. 12. Measure the angles a, b y c, d. FIG. 140. 13. Use your protractor to draw angles of (i) 30, (ii) 90, (iii) 48, (iv) 124, (v) 220, (vi) 300. USE OF INSTRUMENTS 137 14. Measure the angles a, 6, c, d and write down their sum. FIG. 141. 15. Measure the angles a, b and write down their sum. a / 6 FIG. 14'2. 16 Measure the angles a, x 9 6, y. What do you notice about them ? FKJ. 143. 17. Measure the angles AOB, BOG, AOC. 138 CONCISE GEOMETRY 18. Measure the three angles of the triangle ABC and write down their sum. A FIG. 145. 19. Measure the three angles of the triangle DEF and write down their sum. FIG. 146. 20. Without measurement, say which is the larger of the angles, a in Fig. 147 or b in Fig. 148, and roughly by how much. FIG. 147. 21. Draw by eye (with a ruler) angles of 15, 30, 60, 110, 160. Measure them and write down your errors. 22. Without measurement state whether the angles a, 6, c, d, e in Fig. 148 are acute or obtuse or reflex. d FIG. 148. 23. Draw an angle ABC equal to 108; produce CB to D. Measure Z.ABD. USE OF INSTRUMENTS 139 24. Draw an angle AOB equal to 82 ; produce AO, BO, to C, D. Measure /.COD. 25. Draw any five-sided figure ABODE and produce AB, BO, CD, DE, EA. Measure each of the five exterior angles so formed and write down their sum. 26. Draw any triangle ABO; ZCBA, /.CAB, Z.ACD. ,/ ACD? 27. Draw a figure like Fig. 149; find by measurement the values produce BO to D. Measure Is Z.CBA+Z.CAB equal to of Z.ABC+ Z.ADC+ /.BAD and /.BCD. 28. Enlarge Fig. 150, making^ AB = 8 cms., AD = BO = 2 cms., /.DAB = 90= Z.CBA. O is the mid-point of AB. Mark points F, G, H, K, L, M, N on CD such that the lines joining them to O make with OB angles of 30, 50, 70, 90, 110, 130, 150. Measure in cms. FG, GH, HK. O B FIG. 150. USB OF COMPASSES 29. Draw a circle, centre O; draw any diameter AB; take any three points P, Q, R on the circumference. Measure /. s APB, AQB, ARB. 30. Draw two circles of radii 3 cms., 4 cms. so that their centres are 5 cms. apart. Draw their common chord, i.e. the line 140 CONCISE GEOMETRY joining the points at which they cut, and measure its length. What is the angle at which it cuts the line joining the centres 1 31. Take two points A, B 3 cms. apart; construct two points P, Q such that PA = PB -= 5 cms. = QA = QB. 32. Take a point P; describe a circle of radius 4 cms. passing through P; construct a chord PQ of length 6 cms. 33. Draw a circle ; take four points A, B, C, D in order on it. Measure (i) /. ACB and Z.ADB; (ii) Z. ABC and /.ADC. What do you notice 1 34. Draw a large triangle ABC (not isosceles) ; describe circles on AB and AC as diameters. Do they meet on BC 1 35. Take two points A, B 5 cms. apart. Construct a point C such that CA=Gcms, CB = 7 cms. Draw circles with centres A, B, C and radii 2, 3, 4 cms. respectively. What do you notice about them 1 36. Take two points A, B 3 cms. apart. Construct a point C such that CA = CB = 6 cms. Join CA, CB and measure Z.CAB, Z.CBA, Z.ACB. Is L CAB equal to Z.CBA? Is /.CAB equal to twice L ACB '] 37. Draw a circle of radius 3 cms. and place in it 6 chords each of length 3 cms., end to end; what figure is obtained? Measure the angle between two adjacent chords. 38. Draw a straight line AB ; construct a point C such that CA = CB = AB. Measure the angles of A ABC. 39. Draw a straight line AB and take any point P outside it. Construct a point Q such that QA= PA and QB= PB. Join PQ and let it cut AB at R. Measure /. PRA. 40. Draw two circles of radii 3 cms., 4 cms. so that the part of the line joining their centres which lies inside both circles is 1 cm. 41. Draw a line AB 5 cms. long; construct a point C so that CA = 3 cms., CB = 4 cms. Join CA, CB. Bisect with dividers or by measurement AB at D. Measure /. ACB and CD. Is COCAS'! 42. Draw a line AB 3 cms. long ; construct a circle of radius 4 cms. to pass through A and B. USE OF INSTRUMENTS 141 43. Take two points A, B 6 cms. apart. Construct 10 positions of a point P (on either side of AB) such that PA-f PB= 10 cms. (e.g. PA = 3, PB = 7 or PA- 4, PB = 6, etc.). All these positions lie on a smooth curve called an ellipse: draw freehand a curve through these positions. Would you expect the curve to pass through A or B ] 44. Draw a circle, centre O, and take any point T outsile it; on TO as diameter describe a circle cutting the first at P, Q. Join TP, TQ and produce both. What do you notice about these lines i 45. Draw a circle, centre O, of radius 3*5 cms. ; draw a chord PQ such that ^POQ 72. Construct four other chords QR, RS, etc., end to end, each equal to PQ. What is the figure so obtained ? 46. Draw two unequal circles intersecting at P, Q; draw the diameters PX, PY of the circles. Join XY. Does XY pass through Q 1 47. Draw a circle, centre O, and take any six points A, B, C, D, E, F in order on the circumference. Measure /.s ABF, ACF, ADF, AEF, AOF. Do you notice any connection between them? 48. Draw any angle AOB ; with O as centre and any radius (not too short), describe a circle cutting OA, OB at P, Q ; with P, Q as centres and any radius (not too short), de- scribe two equal circles cutting at R. Measure /1AOR, /LBOR. This construction enables you to bisect a given angle. 49. Draw any straight line AB ; with A, B as centres and any radius (not too short), describe two equal circles cutting at P, Q. Join PQ and let it cut AB at R, Measure AR, RB and ^ARP. This construction enables you to draw the perpendicular bisector of a given straight line. 50. Draw any straight line AB and take any point C on it. With C as centre, describe any circle cutting AB at P, Q ; with P, Q as centres and aify radius (not too short), describe two equal circles cutting at R. Join CR. Measure /. ACR. 142 CONCISE GEOMETRY This construction enables you to draw a straight line per- pendicular to a given straight line from a given point on the line. 51. Draw any straight line AB and take any point C outside it. With C as centre, describe any circle cutting AB at P, Q ; with P, Q as centres and any radius (not too short), describe two equal circles cutting at R. Join CR and let it cut AB at S. Measure /. ASC. This construction enables you to draw a straight line per- pendicular to a given straight line from a given point outside tfie line. 52. Draw any straight line AB and take any point C outside it. Take any point P on AB. Join CP and bisect it at Q. With Q as centre and QC as radius, describe 'a circle, cutting AB at R. Join CR. Measure Z.ARC. This construction gives an alternate method to Ex. 51. 53. With any point O as centre, describe a circle ; draw any chord PQ : construct the perpendicular bisector of PQ. Does it pass through O ? 54. Draw a triangle ABC (not isosceles) ; construct the per- pendicular bisectors of AB and AC ; let them meet at O ; with O as centre and OA as radius, describe a circle. Does the circle pass through B and C 1 55. In Fig. 151, without producing AB, construct a line through C perpendicular to AB. xC A B FIG. 151. 56. Draw a line AB, construct a line through B perpendicular to AB without producing AB. 57. Draw an obtuse-angled triangle ABC; construct the per- pendiculars from each vertex to the opposite side. Are they concurrent ? 58. Draw a circle and take four points A, B, C, X on it ; con- struct the perpendiculars XP, XQ, XR to BC, CA, AB. What do you notice about P, Q, R ? USE OF INSTRUMENTS 143 59. Draw a circle of radius 3 cms. and take points A, B, C on it such that AB = 4 cms., AC = 5 cms. Measure Z.BAC: is there more than one answer ? 60. Draw a line AB and take any two points C, D outside it ; construct a point P on AB such that PC= PD. 61. Draw any triangle (not isosceles) and construct the bisectors of its three angles. What do you notice about them ? 62. Draw any triangle ABC; construct the bisectors of /.s ABC, ACB and let them meet at I. Construct the per- pendicular IX from I to BC. With I as centre and IX as radius, describe a circle. What do you notice about this circle 1 63. Draw two lines ABC, BD, cutting at B; construct the bisectors BP, BQ of ZABD, Z.CBD; measure L PBQ. 64. Construct (without using a protractor) angles of (i) 30, (ii) 45, (iii) 105, (iv) 255. 65. Draw a circle and take any three points A, B, C on it (AB:4=AC); construct the perpendicular bisector of BC and the bisector of /. BAG and produce them to meet. What do you notice about their point of intersection ? 66. Draw an obtuse angle and construct lines dividing it into four equal angles. 67. Draw a triangle ABC (not isosceles) ; construct a point P on BC such that the perpendiculars from P to AB and AC are equal. 68. Draw a right angle and construct the lines trisecting it. 69. Draw a line PQ (see Fig. 152), cutting two other lines AB, CD at P, Q; the bisectors of Z.s APQ, CQP meet at H; the bisectors of Z.s BPQ, DQP meet at K; verify that HK when produced passes through the point of intersection of AB and CD and bisects the angle between them. 144 CONCISE GEOMETRY 70. Copy the following figures 153-167 on any convenient scale. 153 154 155 156 165 166 167 FIGS. 153-167 USE OF SET SQUARES 71. Draw, a line AB and take three points P, Q, R outside it : use set squares to draw lines through P, Q, R parallel to AB. 72. Draw a line AB and take three points P, Q, R outside it : use set squares to draw lineo through P, Q, R perpendicular to AB. DRAWING TO SCALE 145 73. Draw a line AB and ta.e <* point C on it : use set squares to draw a line through C perpendicular to AB. 74. Draw a line AB and take a point P outside it : use set squares to draw two lines PQ, PR making angles of 60 with AB. 75. Draw a triangle ABC and use set squares to draw its three altitudes (i.e. perpendiculars from each corner to the opposite side). 76. Draw a triangle ABC and use set squares to complete the parallelogram ABCD : measure its sides. 77. Use set squares to draw a four-sided figure having its opposite sides parallel and one angle a right angle : measure the diagonals. 78. Draw a triangle ABC (not isosceles) ; bisect AB at H ; use set squares to draw a line HK parallel to BC to meet AC at K ; measure AK, KC. 79. Draw any angle BAC and cut off AB equal to AC ; use set squares to construct bisector of /. BAC. 80. Use set squares to draw a right angle, and use them to trisect it. 81. Draw a line AB and divide it into five equal parts as follows : draw any other line AC and cut oft' from AC five equal parts AP P Q> Q R > RS > ST ; J " 1 BT J through P, Q, R, S draw lines parallel to TB : these cut AB at the required points. DRAWING TO SCALE EXEBCISE XXVII FIG. 168. P R FIG. 169. DEFINITIONS. (i) In Fig. 168, if OA is horizontal, Z.AOB is called the angle of elevation of B as viewed from O. (ii) In Fig. 169, if QH is horizontal, Z.HQR is called the angle of depression of R as viewed from Q. 1. A courtyard is 80 feet long and 50 feet wide ; what is the distance between two opposite corners 1 10 146 CONCISE GEOMETRY 2. A gun whose range is 5000 yards is in position at a point 3500 yards from a straight railway line; what length of the line can it command 1 3. A ladder, 1 5 feet long, is resting against a vertical wall ; the foot of the ladder is 6 feet from the wall ; how high up the wall does it reach ? 4. The ends of a cord, 10 feet long, are fastened to two nails each of which is 15 feet above the ground; the nails are 5 feet apart ; a weight is attached to the mid-point of the cord : how high is it above the ground 1 5. A straight passage runs from A tq B, then turns through an angle of 70 and runs on to C ; if AB is 80 yards and BC is 100 yards, what distance is saved by having a passage direct from A to C 1 6. A man rows due north at 4 miles an hour, and the current takes him north-east at 5 miles an hour ; how far is he from his starting-point after 20 minutes 1 7. A man starts from A and walks 2 miles due south to B, then 3 miles south-west to C, then 1 mile west to D ; what is the direction and distance of D from A ? 8. Southampton is 12 miles S.S.W. of Winchester; Romsey is 10 miles W. 32 S. of Winchester. Find the distance and bearing of Romsey from Southampton. 9. An aeroplane points due north and flies at GO miles an hour ; the wind carries it S.W. at 15 miles an hour. What is its position ten minutes after leaving the aerodrome ? 10. Andover is 12 miles from Winchester and 15 miles from Salisbury ; Salisbury is 20 miles W. of Winchester. [Andover is north of the Salisbury- Winchester line.] Find the bearing of Andover from Salisbury. 11. Exeter is 42 miles from Dorchester and 64 miles from Bristol ; Bristol is 55 miles due north of Dorchester; Barnstable is 33 miles N.E. of Exeter. What is the distance and bearing of Barnstable from Dorchester ? 12. A weight is slung by two ropes of lengths 12 feet, 16 feet, from two pegs 18 feet apart in a horizontal line. What is the depth of the weight below the line of the pegs) DRAWING TO SCALE 147 13. From two points 500 yards apart on a straight road running due north, the bearings of a house are found to bo N. 40 E. and E. 20 S. ; find the shortest distance of the house from the road. 14. There are two paths inclined at an angle of 40 which lead from a gate across a circular field : one ruiif across the centre of the field and is 120 yards long; what is the length of the other ? 15. A path runs round the edge of aequare ploughed field ABOD ; if you follow the path from A to C yot go 50 yards farther than if you walk straight across. What is the length of a side of the field ? 16. One end of a string, 5 feet long, is fastened to a nail, and a weight is attached to the other end; the weight swings backwards and forwards through 15 each side of the vertical. What is the distance between its two extreme positions ? 17. At a distance of 40 yards from a tower, the angle of elevation of the top of the tower is 35 ; find the height of the tower in feet. 18. A kite is flown at the end of a string 120 yards long which makes an angle of 65 with the ground: find in feet the height of the kite. 19. What is the elevation of the sun when a pole 12 feet high casts a shadow 20 feet long t 20. A fenced level road running due north suddenly turns due east, with the result that the shadow of the fence is increased in breadth from 3 feet to 5 feet: what is the bearing of the sun f 21. The elevation of the top of a chimney is 20; from a place 60 yards nearer, it is 30 ; find its height in feet. 22. From the top of a cliff 150 feet high, the angle of depression of a boat out at sea is 20 ; what is the distance of the boat from the cliff in yards t 23. From the top of a tower 250 feet high, the angles of depression of two houses in a line with and at the same level as the foot of the tower are 61 and 48. Find their distance apart in yards. 148 CONCISE GEOMETRY MISCELLANEOUS CONSTRUCTIONS I EXERCISE XXVIII 1. Draw an angle BAG and a line PQ; construct points R, S on AB, AC such that RS is equal and parallel to PQ. 2. Draw a circle and construct points P, Q, R on it such that PQ = QR=RP; take any other point X on the circle. Measure XP, XQ, XR and verify that the longest of these equals the sum of the other two. 3. Draw an angle BAG of 50 ; construct on AB, AC points P, Q such that Z.QPA = 90 and PQ = 4 cms. Measure AP. 4. Draw a circle of radius 4 cms., and take a point A at a distance of 2*5 cms. from the centre: construct a chord PQ passing through A and bisected at A. 5. Draw a large quadrilateral ABCD, so that AB is not parallel to CD ; construct a point P such that PA = PB and PC = PD 6. Draw a line AB and take a point C distant 2 // from AB ; construct a circle with C as centre, cutting AB at two points S" apart. Measure its radius. 7. Draw an angle BAG of 70; construct a point P whose distances from AB, AC are 3 cms., 4 cms. Measure AP. S. Draw a line AB and take a point C distant 2" from AB ; construct two points P, Q each of which is 1* from AB and 1 " from C. Measure PQ. 9. Draw two lines AB, AC and take a point P somewhere between them ; construct a line to pass through P and cut oft' equal lengths from AB and AC. 10. Draw two lines AB, CD and take any point E between them. Construct a line to pass through E and the (inaccessible) point of intersection of AB, CD. [Use the system of parallel lines shown in Fig. 170.] B FIG. 170. MISCELLANEOUS CONSTRUCTIONS 1 149 11. Draw a triangle ABC; construct a line through C parallel to the bisector of /. BAG and let it meet BA produced at E. Measure AE, AC. 12. Draw a circle and take two points A, B outside it. Construct a circle to pass through A, B and have Its centre on tho first circle. When is this impossible ? 13. Draw a circle and take a point H 01 tside it \ draw two lines HAB, HOC, cutting the circle at A, B, D, C ; join AD, BC, and produce them to meet at K. Construct a circle to pass through H, A, D and a second circle to ;*ass through K, D, C. Do these circles cut again at a point on HK 1 14. Construct five points in the same relative position to each other as are A, B, C, D, E in Fig. 171. FIG. 171. 15. Take a line AB and a point C outside it such that the foot of the perpendicular from C to AB would be off the page. Construct that portion of the perpendicular which comes on the page. 16. Take a line AB and a point C and suppose there is an obstacle between C and AB which a set square cannot move over (see Fig. 172). Construct a line through C parallel to AB. B FIG. 172. 17. By folding, obtain a crease* which (i) bisects a given angle, (ii) bisects a given line at right angles. 150 CONCISE GEOMETRY 18. By folding, obtain the perpendicular to a given line from a given point outside it. 19. By folding, obtain an angle of 45. 20. Take a triangular sheet of paper and find by folding the point which is equidistant from the three corners. CONSTRUCTION OF TRIANGLES, PARALLELOGRAMS, ETC. EXERCISE XXIX 1. Construct, when possible^ the triangle ABC from the following measurements, choosing your own unit. If there are two different solutions, construct both : a Fio. 173. (i) a = 3, 6 = 4, c = 5, measure A. (ii) a = 3, 6 = 4, c = 8, measure A. (iii) a = 5, B = 30, C = 45, measure 6. (iv) a = 4, A =48, B = 33, measure h. (v) a =7, A=110, B = 40, measure 6. (vi) a = 5, B=125, C = 70, measure 6. (vii) 6 = 5, c = 7, C = 72, measure a. (viii) 6 = 6, c = 4, C = 40, measure a. (ix) 6 = 8, c=6, C=65, measure a, (x) A = 40, B = 60, C = 80, measure a. (xi) A = 50, B = 40, C = 70, measure a. (xii) A =125, 6=7-3, c = 5'4, measure a. (xiii) A = 90, a= 11-2, 6=7-3, measure c. (xiv) a = 6 = 6*9, A = 50, measure c. (xv) a =26, 0=^-, measure A. Jtt 2. Draw two unequal lines AC, BD bisecting each other; join AB, BC, CD, DA and measure them. ABCD is a parallelogram. CONSTRUCTION OF TRIANGLES 151 3. Draw two equal lines AC, BD bisecting each other; join AB, BC, CD, DA; measure /.ABC. ABCD is a rectangle. 4. Draw two unequal lines AC, BD bisecting each other at right angles; join AB, BC, CD, DA and Treasure them. Af>CC is a rhombus. 5. Draw two equal lines AC, BD bisecting each ot^er at right angles ; join AB, BC, CD, DA ; measure AB, BC, /. ABC. ABCD is a square. f>. Draw two unequal perpendicular lines AC, BD such that AC bisects BD; join AB, BC, CD, DA and measure them. ABCD is a kite. 7. Draw an angle of 57 and cut off AB, AC from the arms of the angle so that AB = 5 cms., AC = 8 cms. ; construct a point D such that BD = AC and CD^ AB. What sort of a quadrilateral is ABCD ? 8. Construct a parallelogram ABCD, given AB = 7 cms., AC = 10 cms., BD = 8 cms. ; measure BC, CD. 9. Construct an isosceles triangle with a base of 6 cms. and a vertical angle of 70 ; measure its sides. 10. Construct a rhombus ABCD, given AB- 5 cms., AC~G cms. ; measure /.BAD. 11. Construct an isosceles triangle of base 4*6 cms. and height 5 cms. ; measure its vertical angle. 12. Construct the quadrilateral ABCD, given AB=BC=3 cms., AD = DC = 5 cms., /_ ABC = 1 20 ; measure /. ADC. 13. Construct the rhombus ABCD, given AC = 6 cms., BD = 9 cms. ; measure AB. 14. Construct the rhombus ABCD, given L ABC = 40, BD = 7 cms. ; measure AC. 15. Construct a rectangle ABCD, given BD = 8 cms. and that AC makes an angle of 54 with BD ; measure AB, BC. 16. Construct a trapezium ABCD with AB, CD its parallel sides such that AB - 8, BC - 4, CD - 3, AD - 2 ; measure L BAD. 17. Construct the quadrilateral ABCD, given that (i) AB = 4, BC-4-5, CD -3, /.ABC -80, Z.BCD = 110; measure AD; (ii) AB-5, AC = 6, AD = 4, BD-7, CD = 3 ; measure BC, 152 CONCISE GEOMETRY (iii) L ABC = 70, L BCD = 95, ^CDA=105, AB = 5, AD = 4 ; measure BC. (iv) AB = 5, BC = 6, CD = 3, DA=4'S, Z.ADC=100; measure /.ABC. (v) AB = 5, /. CAB = 35, ,/ ABD = 47, Z.ACB-65 , /.ADB = 54 ; measure CD. 18. Construct the triangle ABC, given that (i) a-f &= 11, 1> + c= 16, c4-a = 13 ; measure A. (ii) A - B = 25, C = 55, c = 7 ; measure a. (iii) A : B : C = 1 : 2 : 3, a = 3 j measure c. (iv) A + B = ]18, B + C = 96, a==7; measure c. 19. Construct an equilateral triangle ABC such that if D is a point on BC given by BD = 3 cms., then L DAC = 40 ; measure BC. 20. Construct a square having one diagonal 5 cms. ; measure its side. 21. AD is an altitude of the triangle ABC; given AD = 4 cms., L ABC = 55, /. ACB = 65, construct A ABC; measure BC. 22. AE is a median of the triangle ABC; given AB = 4 cms., AC = 7 cms., AE=4*5 cms., construct A ABC; measure BC. 23. AD is an altitude of the triangle ABC; given AB = 6 cms. AD = 4 cms., /.ACB = 68, construct A ABC; measure BC. 24. AD is an altitude of A ABC; AD = 4 cms., L BAG = 75, /.ABC = 50, construct A ABC; measure BC. 25. The distances between the opposite sides of a parallelogram are 3 cms., 4 cms., and one angle is 70 ; construct the parallelogram and measure one of the longer sides. 26. Construct a parallelogram of height 4 cms., having its diagonals 5 cms., 8 cms. in length : measure one of the longer sides. 27. Construct an equilateral triangle of height 4 cms. ; measure its side. 28. Construct the triangle ABC, given that (i) a + J = 2c=14, A =70; measure a. (ii) a + 6 + c=20, A = 65, B = 70; measure a. (iii) a = 10, 6-fc=13, A = 80; measure b. (iv) a = 8, & + c=10, B = 35; measured. (v) a = 9, c-i = 4, B = 25; measure c. (vi) a= 9, b - c= 2, A = 70 ; l measure i. (vii) a = 5, 6 = 3, A - B = 20 ; measure c. MISCELLANEOUS CONSTRUCTIONS II 153 29. Construct an isosceles triangle of height 5 cms. and perimeter 18 cms. ; measure its base. 30. Each of the base angles of an isosceles triangle exceeds thy vertical angle by 24 ; the base is 4 cms. ; construct the triangle and measure its other sides. MISCELLANEOUS CONSTRUCTIONS II EXERCISE XXX 1. Given two points H, K on the same side of a given line AB, construct a point P on AB such that PH, PK make equal angles with AB. 2. Given two points H, K on opposite sides of a given line CD, (see Fig. 174), construct a point P on CD such that Z.HPC- L KPC. *K Fia. 174. 3. Given a triangle ABC, construct a line passing through A from which B and C are equidistant. 4. Given a triangle ABC, construct a line parallel to BC, cutting AB, AC at H, K such that BH + CK= HK. 5. Given a square ABCD, construct points P, Q on BC, CD such that APQ is an equilateral triangle. 6. Given a triangle ABC, construct a rhombus with two sides along AB, AC and one vertex on BC. 7. Given two parallel lines AB, CD and a point P between them, construct a line through P, cutting AB, CD at Q, R such that QR is of given length. 8. Given a triangle ABC, construct a point which is equidistant from B and C and also equidistant from the lines AB and AC. 154 CONCISE GEOMETRY 9. Given in position the internal bisectors of the angles of a triangle and the position of one vortex, construct the triangle. 10. By construction and measurement, find the height of a regular tetrahedron, each edge of which is 2*. 11. A room is 20 feet long, 15 feet wide, 10 feet high ; a cord is stretched from one corner of the floor to the opi>osite corner of the ceiling, find by drawing and measurement the angle which the cord makes with the floor. 12. Construct a square such that the length of its diagonal exceeds the length of its side by a given length. EXAMPLES ON THE CONSTRUCTIONS OF BOOK II AREAS EXERCISE XXXI 1. Find the areas of the following figures, making any necessary constructions and measurements : (i) A ABC, given 6-5, <? = 4, A = 90. (ii) Rectangle ABCD, given AB = 7, AC= 10. (iii) A ABC, given a = 5, 6=6, c=7. (iv) A ABC, given b = 5, c = 4, B = 90. (v) A ABC, given 6 = c= 10, a= 12. (vi) A ABC, given a =6, B=130, C = 20. (vii) Hgrani ABCD, given AB = 8, AD = 6, Z.ABC -70. (viii) A rhombus whose diagonals are 7, 8. (ix) A trapezium ABCD, given AB = 5, BC = 6, CD = 9, L BCD = 30, and AB parallel to DC. (x) Quad. ABCD, given AB = 3, BC=5, CD = 6, DA -4, BD = 5. 2. Draw a triangle whose sides are 5, 6, 8 cms. and obtain its area in three different ways. 3. Draw a triangle with sides 5, 6, 7 cms., and construct an isosceles triangle with base 6 cms. equal in area to it ; measure its sides. 4. Construct a parallelogram of area 21 sq. cms. such that one side is 6 cms., one angle is 50 ; measure the other side. 5. Construct a parallelogram of area 15 sq. cms. with sides 5 cms., 6 cms. ; measure its acute angle. 6 Draw a triangle with sides 4, 5, 6 cms., and construct a parallelogram equal in area to it and having one side equal 155 156 CONCISE GEOMETRY to 4 cms. and one angle equal to 70; measure the other side. 7. Construct a rhombus each side of which is 5 cms. and of area 15 sq. cms. ; measure its acute angle. 8. Draw a parallelogram with sides 4 cms., 6 cms., and one angle 70; construct a parallelogram of equal area with sides 5 cms., 7 cms. ; measure its acute angle. 9. Construct a parallelogram of area 20 sq. cms., with one side 5 cms., and one diagonal 7 cms. ; measure the other side. 10. Draw a triangle with sides 5, 6, 8 cms., and construct a triangle of equal area with sides 5*5, 6*5 cms.; measure the third side. 11. Construct a parallelogram equal in area to a given rectangle and having its sides of given length. 12. Construct a triangle equal in area to a given triangle and having one side equal in length to a given line, and one angle adjacent to that side equal to a given angle. 13. Draw a quadrilateral ABCD such that AB = 6 cms., BC = 5 cms., CD = 4 cms., L ABC =110, L BCD = 95. Reduce it to an equivalent triangle with AB as base and its vertex on BC. Find its area. 14. Draw a figure like Fig. 149, and reduce it to an equivalent triangle having AB as base and its vertex on AD. A FIG. 149. 15. Draw a figure like Fig. 175 and reduce it to an equivalent triangle. FIG. 175. AREAS 157 16. Given four points A, B, C, D as in Fig. 176, construct a point P such that the figures ABPD and ABCD are of equal area and DP is i>erpendicular to AB. C FIG. 176. 17. Given a parallelogram ABCD and a point O inside it, construct a line through O which divides ABCD into two parts of equal area. 18. Given a triangle ABC and a point D on BC such that BD< jBC, construct a point P on AC such that (i) ADPC = J A ABC, (ii) ADPC = f AABC. 19. Given a parallelogram ABCD, construct point P, Q on BC, CD such that AP, AQ divide the parallelogram into three portions of equal area. 20. Given a quadrilateral ABCD, construct a line through A which divides the quadrilateral into two parts of equal area. 21. Given a quadrilateral, construct lines through one vertex which divide it into five parts of equal area. If ABCD is any parallelogram, and if P is any point on BD, and if lines are drawn through P parallel to AB, BC as in Fig. 177, the parallelograms AP, PC are of equal area. Use this fact for the following construction : Construct a parallelogram equal in area to and equiangular to a given parallelogram and having one side of given length. Fio. 177. 158 CONCISE GEOMETRY 23. Given a triangle ABC, construct a point G inside it such that the triangles GAB, GBC, GCA are of equal area. 24. Given a quadrilateral ABCD, perform the following con- struction for a line BP bisecting it (see Fig. 178). Bisect AC at O ; through O draw OP parallel to BD to meet CD (or AD) at P join BP. C SUBL i .IJSION OF A LINE EXERCISE XXXII 1. Draw a line ABj;| divide into three equal parts without measuring it. 2. Draw a line AB and construct a point P on AB such that AP a = 4. PB * 3. Draw a line AB and construct a point Q on AB produced, such -g-J- 4. Divide a given line in the ratio 5 : 3 both internally and externally. 5. Construct a diagonal scale which can be used for measuring lengths to ^ inch. 6. By using a diagonal scale, draw a line of length 2*73 inches : on this line as base construct an isosceles right-angled triangle and measure its equal sides as accurately as possible. 7. Use a diagonal scale to measure the hypotenuse of a right- angled triangle whose sides are 2* and 3*. 8. On a scale of 6" to the mile, what length represents 2000 yards 1 Draw a scale showing hundreds of yards. SUBDIVISION OF A LINE 159 9. What is the R.F. [i.e. representative fraction] for a map of scale 2" to the mile 1 Construct a scale for reading off distances up to 5000 yards, and as small as 500 yards 10. The It.F. of a map is 1 : 20,000 ; express this in inches to the mile and Construct a suitable SCF ta 'o read miles ana tenths of miles. 11. Given two lines AB, AC and a poii c P between them, construct a line through P, cutting AB, AC at Q, R so that QP= PR. 12. Given two lines AB, AC and a point P between them, construct a line through P with its extremitie on AB, AC and divided at P in the ratio 2:3. 13. Draw a triangle ABC such that BC=^6 cms. ; construct a line parallel to BC, cutting AB, AC at H, K such that HK = 2 cms. What is the ratio AH : HB ? 14. Given a triangle ABC, construct a line parallel to BC, cutting AB, AC at H, K such that HK=BC. EXAMPLES ON THE CCNS1 RATIONS OF BOOK III CONSTRUCTION OF CIRC) &S, ETC. EXERCISE XXXIII 1. Use a coin to draw a circle, and construct its centre. 2. Given two points A, B and a line CD, construct a circle to pass through A and B and have its centre on CD. 3. Draw a line AB 3 cms. long, and construct a circle of radius 5 cms. to pass through A and B. 4. Draw two lineb AOB, COD intersecting at an angle of 80 ; make AO = 3 cms., OB = 4 cms., CO = 5 cms., OD = 2-4 cms. ; construct a circle to pass through A, B, C. Does it pass through D ? 5. Construct two circles of radii 4 cms., 5 cms., such that their common chord is of length 6 cms. Measure the distance between their centres. 6. Draw two lines OAB, OCD intersecting at an angle of 40 ; make OA=2 cms., OB = 6 cms., OC 3 cms., OD = 4 cms. ; construct a circle to pass through A, B, C. Does it pass through D ? 7. Given a circle and two points A, B inside it, construct a circle to pass through A and B and have its centre on the given circle. 8. Given a point B 'on a given line ABC and a point D outside the line, construct a circle to pass through D and to touch AC at B. 9. Draw a line AB and take a point C at a distance of 3 cms. from the line AB; construct a circle of radius 4 cms. to pass through C and touch AB. 10. Draw two lines AB, AC making an angle of 65 with each other , construct a circle of radius 3 cms. to touch AB and AC. n 162 CONCISE GEOMETRY 11. Draw a circle of radius 3 cms. and take a point A at a distance of 4 cms. from its centre ; construct a circle to touch the first circle and to pass through A, and to have a radius of 2 cms. Is there more than one such circle t 12. Given a straight line and a circle, construct a circle of given radius to touch both the straight line and the circle. Is this always possible 1 ? If not, state the conditions under which it is impossible. 1 3. Draw a line AB of length 6 cms. : with A, B as centres and radii 3 cms., 2 cms. respectively, describe circles. Construct a circle to touch each of these circles and have a radius of 5 cms. Give all possible solutions. [The contacts may be internal or external.] 14. Draw a circle of radius 4*5 cms., and draw a diameter AB; construct a circle of radius 1-5 cm. to touch the circle and AB. 15. Given a circle and a point A on the circle and a point B outside the circle, construct a circle to pass through B and to touch the given circle at A. 16. Draw a circle of radius 5 cms.; construct two circles of radii 1'5 cm., 2*5 cms. touching each other externally and touching the first circle internally. 17. Draw a triangle whose sides are of lengths 2, 3, 4 cms., and construct the four circles which touch the sides of this triangle and measure their radii. 18. Draw two lines OA, OB such that Z. AOB = 40, and OA = 4 cms. ; construct a circle touching OA at A and touching OB ; measure its radius. 19. Given a triangle ABC, construct a circle to touch AB, AC and have its centre on BC. Is there more than one solution ? 20. Inscribe a circle in a given sector of a circle, [i.e. Given two radii OA, OB of a circle, construct a circle to touch OA, OB and the arc AB.] 21. Given two radii OA, OB of a circle, construct points H, K on OA, OB such that the circle on HK as diameter touches the arc AB. 22. Given two points A, B and a point D on a line CDE, construct two concentric circles one of which passes through A, B and the other touches CE at D. When ia this impossible ? CONSTRUCTION OF CIRCLES, ETC. 163 23. Given three points A, B, C, construct three circles with points as centres so that each circle touches the other tw>. Is there more than one solution ? 24. Draw two lines OA, OB intersecting at an angle of 40; construct a circle touching OA an. 1 OB and auoh tua u tne chord of contact is of length 4 cms. ; mea UTJ i+* radius. 25. Inscribe a circle in a given rhombus 2fi. Q ; ven two points A, B, 4 cms. apait, construct a circle to pass through A and B and such that the tangents at A and B include an angle of 100 ; measure 's radius. 27. Find by measurement the radius of the circle inscribed in the triangle whose sides are of lengths 6, 7, > cms. 28. ABC is a triangle such that BC - 6 cms., BA = 4 cms., L ABC = 90 ; find by measurement the radius of the circle escribed to BC. 29. Given two parallel lines and a point between them, construct a circle to touch the given lines and pass through the given point. 30. Draw a quadrilateral so that its sides in order are 4, 5, 7, 6 cms. ; inscribe a circle in it to touch three of the sides. Does it touch the fourth side ? 31. In Fig. 179, AB, CD are two given parallel lines: construct a circle to touch AB, CD and the given circle. C D FIG. 179. 32. Given two parallel lines AB, CD and a circle between them, construct a circle to touch AB, CD and to touch and enclose the given circle. 33. Given two circles, centres A, B, radii a, &, and a point C on the first, construct a circle to touch the first circle at C and also to touch the second. Fig. 180 gives the construction for the centre P of the required circle, if it touches both circles externally. D is found by making CD = b. Perform 16 CONCISE GEOMETRY this construction and construct also the circle in the case where the contacts are external with circle A, internal with circle B. How would C be situated if the constructed circle touches circle A internally and circle B either internally or externally ? Fm. 180. 34. ABC is an equilateral triangle ; AB = 4 cms. ; A, B are the centres of two equal circles of radii 2*5 cms. ; CA is produced to meet the first circle at D. Construct a circle touching the first circle internally at D and touching the second circle externally. State your construction. 35. Construct a circle to touch a given line AB and a given circle centre C, at a given point D. Fig. 181 gives the construc- tion for the centre P of the required circle if the contact is external. Perform the construction and construct the case where the contact is internal. Fio. ,181. CONSTRUCTION OF CIRCLES, ETC. 165 Construct the Figs, in exs. 36-62 : do not rub out any of your construction lines. 36. Three arcs each of radius 3 cms. and each Jth of a complete circumference. A B<~ FIG. 182. 37. AB, BC, CD, DE are equal quadrants; AE ~ fi cms. FIG. 183. 38. AB, BC, CD, DE, EF, FG, GH, HA are alternately semicircles and quadrants of equal radius ; XY= 10 cms. y FIG. 184. 39. Three arcs each of radius 3 cms. touching at A, B, C. FIG. 185. 166 CONCISE GEOMETRY 40. The aides of the rectangle are 6 cms., 8 cms. Fio. 186. 41. The radii of the arcs AB, BC, CA are 3 '5 cms., 2 '5 cms., 7 cms. FIG. 187. 42. The radii of the circles are 1 cm., 2 cms., 2 cms., 3 cms., and the centre of the smallest circle lies on the largest. FIG. 188. 43. AP, AQ are arcs of radii 4 cms. ; PQ is of radius 8 cms. ; AB is perpendicular to CD and equals 3 cms. B Fio. 189. CONSTRUCTION OF CIRCLES, ETC. 167 44. AB, BC, CD arc arcs of radii 3 cms., AD equals 7 cms. and touches AB, DC. 45. The radii of the arcs AB, BC are 3*5 cms., 1*2 cm., CD = 5 cms., DE = 6'5 cms., AE = 7 cms. 90 90 C D FIG. 191. 46. AB, AD are arcs of radii 6 cms. ; AC equals 6 cms. and is an axis of symmetry. 168 CONCISE GEOMETRY 47. BC is a quadrant ; the radii of arcs AB, BC, CA are 4, 2, 3 cms. D Fm. 193. 48. AF is an axis of symmetry; AB, BC, DE are equal quadrants; AF = 8 cms., EG = 6 cms. E F G 49. The radii of the arcs ABC, ADC are 3, 5'5 cms. ; chord AC = 5 cms. Construct the figure and inscribe in it a circle of radius 1*5 cm. A 50. CE is an axis of symmetry ; AB, BC are arcs each of radius 3 cms.; the centre of AB lies on AD. AD = 10 cms., CE = 5 cms. C F FIG. 196. CONSTRUCTION OF CIRCLES, ETC. 169 51. AB is an arc of radius 3 cms. ; BC, CD, DA are arcs each of radius 1 cm. ; chord AE = chord EB = 3 cms. 52. AB, BC are semicircles, each of radius 2 cms. intersecting at an angle of 120. The arc AC touches ar< ^ AB, CB. 53. AB, DE are arcs each of radius 2 cms. with their centres on AE ; BC = CD = 4 cms. ; AE = 6 cms. FIG. 199. 54. CD is an axis of symmetry; AB = 9'5 cms., CD = 3 '5 cms. ; AE, EC are arcs of radii 2, 10 '5 cms. respectively. 170 CONCISE GEOMETRY 55. AB is a quadrant of radius 2 -5 cms. with its centre on AC; AC = 7 cms. The arc BC touches AB at B. Fiu. 201. 56. ABCD is a square of side 2 cms. ; BE, EF are circular arcs with C, A as centres respectively. B A FIG. 202. 57. AB is an axis of symmetry; PAQ is a semicircle of radius 2 cms. ; RBS is an arc of radius 1 cin. ; AB = 7 cms. The arcs PR, QS are tangential at each end. A 58. AB = 3'5 cms., AC = 6 cms., Z.BAC = 90; radius of arc CP is 1*5 cm. Bl A FIG. 204. CONSTRUCTION OF CIRCLES, ETC. 171 59. AB=BC=3 cms.; tho arcs AB, BC cut the line ABC at angles of 30. 60. AB is a semicircle, radius 3 cms., centre O ; O T3 , OQ are arcs each of radius 1 cm. ; the arcs AP, AB a-e tangential at A. Q/ FIG. 206. 61. Fig. 207 is formed by parts of nine equal circles touching where they meet ; AX, BY, CZ are each axes of symmetry ; the radius of each arc is 1*5 cm. FIG. 207. 62. AB, CD, EF, GH are the diameters of semicircles each of radius 1 cm. and when produced form a square ; AD, BG, CF, HE are arcs each of radius 5 cms. 172 CONCISE GEOMETRY MISCELLANEOUS CONSTRUCTIONS III EXERCISE XXXIV 1. Draw a circle of radius 3 cms., and construct a chord of the circle of length 5 cms. Take a point A inside the circle but not on the chord, and construct a chord of length 5 cms. passing through A. 2. Given a chord PQ of a given circle and a point R on PQ, construct a chord through R equal to PQ. 3. Inscribe a regular hexagon in a given circle. 4. Inscribe an equilateral triangle in a given circle. 5. A, B, C are three given points on a given circle ; construct a chord of the circle equal to AB and parallel to the tangent at C. 6. Draw a circle radius 4 cms. and take a point 6 cms. from the centre. Construct the tangents from this point to the circle and measure their lengths. 7. Draw a circle of radius 3 cms., and construct two tangents which include an angle of 100. 8. Draw a line AB of length 7 cms. ; construct a line AP such that the perpendicular from B to A P is 5 cms. 9. Draw a circle, centre O, radius 4 cms. ; take a point A 6 cms. from O ; draw AB perpendicular to AO ; construct a point P on AB such that the tangent from P to the circle is of length 5*5 cms. ; measure A P. 10. Draw a circle of radius 3 cms. and take a point 5 cms. from the centre ; construct a chord of the circle of length 4 cms. which when produced passes through this point. 11. Draw a line AB of length 5 cms. and describe a circle with AB as diameter ; construct a point on AB produced such that the tangent from it to the circle is of length 3 cms. 12. Given a circle and a straight line, construct a point on the line such that the tangents from it to the circle contain an angle equal to a given angle. 13. Circumscribe an equilateral triangle about a given circle. 14. On a line of length 5 cms., construct a segment of a circle containing an angle of 70 ; measure its radius. MISCELLANEOUS CONSTRUCTIONS III 173 15. On a line of length 2 inches, construct a segment of a circle containing an angle of 140 ; measure its radius. 16. In a circle of radius 3 cms., inscribe a triangle whose angles are 40, 65, 75 ; measure its longest side. 17. Inscribe in a circle of radius V a rectangle of length 1 5", and measure its breadth. 18. Circumscribe about a circle of radius 2 cms. a triangle whose angles are 50, 55, 75 ; measure its longest side. 19. Given three non-collinear points A, B, C, construct the tangent at A to the circle which passes through A, B, C without either drawing the circle or constructing its centre. 20. Draw two circles of radii 2 cms., 3 cms., with their centres 6*5 cms. apart; construct their four common tangents. 21. Draw two circles of radii 2*5 cms., 3*5 cms., touching each other externally, and construct their exterior common tangents. 22. Draw a line AB of length 6 cms. and construct a line PQ such that the perpendiculars to it from A, B are of lengths 2 cms., 4 cms. 23. Draw two circles of radii 2 cms., 3 cms., with their centres 6 cms. apart ; construct a chord of the larger circle of length 4 cms. which when produced touches the smaller circle. 24. Construct the triangle ABC, given that BC = 6 cms., L BAC = 90, the altitude AD = 2 cms. ; measure AB, AC. 25. Construct the triangle ABC, given that BC = 5 cms., Z.BAC = 55, the altitude AD = 4 cms. ; measure AB, AC. 26. Construct the triangle ABC, given the length of BC and the altitude BE and the angle BAC. 27. Construct a triangle given its base and vertical angle and the length of the median through the vertex. 28. Construct a triangle ABC, given BC = 6 cms., L BAG =52, and the median BE = 5 cms. 29. Draw a circle of radius 3 cms., and construct points A, B, C on the circumference such that BC = 5 cms., BA + AC = 8*1 cms. ; measure BA and AC. 30. Draw a circle of radius 3 5 t cms. and inscribe in it a triangle ABC such that BC=5'8 cms., BA- AC = 2 cms. ; measure BA and AC. 174 CONCISE GEOMETRY 31. Construct a triangle ABC given its perimeter, the angle BAC and the length of tha altitude AD. 32. Draw any circle and take two points A, B on it and a point C outside the circle; construct a point P on the circle such that PC bisects Z.APB. 33. Draw two lines which meet at a point off your paper ; construct the bisector of the angle between them. 34*. Draw any triangle ABC (not right-angled). Construct a square PQRS such that PQ passes through A, QR passes through B, and PR cuts QS at C. 35*. Construct the quadrilateral A BCD, given that AD = 5 cms., BC = 4'6 cms., L ABD = Z. ACD = 55, CBD = 43; measure CD. 36*. Draw any circle and take two points A, B on it ; construct a point P on the circle such that chord PA equals twice chord PB. 37*. Draw a circle of radius 3 cms., centre O, and take a point P at distance of 5 cms. from O ; construct a line through P, cutting the circle at Q, R such that the segment QR contains an angle of 70 ; measure /. OPQ. 38*. Draw two unequal circles intersecting at A, B ; construct a line through A, cutting the circles at P, Q such that PA = AQ. 39*. Draw two unequal circles intersecting at A, B ; construct a line through A, cutting the circles at P, Q such that PQ is of given length. 40*. Circumscribe a square about a given quadrilateral. EXAMPLES ON THE CONSTRUCTIONS OF BOOK IV PROPORTION AND SIMILAIl FIGURES EXERCISE XXXV 1. Construct and measure a fourth proportional to lines of length 4, 5, 6 cms. 2. Construct and measure a third proportional to lines of length 5, 6 cms. 3. Draw a line AB and divide it internally in the ratio 2 : 3. 4. Draw a line AB and divide it externally (i) in the ratio 5:3; (ii) in the ratio 3 : 5. 5. Draw a line AB and divide it internally and externally in the ratio 3 : 7. x 7 6. Use a construction to solve - = --. j 5 7. Find graphically the value of (i) 2 - * 'f ; (ii) 3*8 x 2'7. . 8. Construct a line of length y cms. 9. Draw a line AB and divide it in the ratio 2:7:3. 10. Draw any triangle ABC and any line PQ ; construct a triangle such that its perimeter equals PQ and its sides are in the ratio AB : BC : CA. 11. To construct the expressions (i) -, (ii) , proceed as follows : Draw two lines OH, OK (see Fig. 209). From OH, cut off OA = a. From OK, cut off OB = 6, OC = 0, OF-/, OG=#. Join AF, draw BX parallel to FA, cutting OH at X, then a b --. 176 176 CONCISE GEOMETRY Join XG, draw CY parallel to GX, cutting OH at Y, then QY ^= a ^ c a ^ c ~7* </~~ fi Use this construction to find (i) 5 - *- 3 ' 8 , (ii)-L* *"? * Jl?_ 4'7 4*i x I'o and extend it to find , where a, 6, c, d, e y /, </, h are given lengths. O C B F G FIG. 209. 12. If a, 6, r, d are given numbers, construct, by the method of ex. 11, Kg. 209, (i) f *; (ii) ^ ; (iii) g. 13. Given two lines AB, AC and a point D between them, construct a line through D, cutting AB, AC at P,' Q such that PD = DQ. 14. Draw a line ABCD; if AB = # cms., BC-y cms., CD = 2 cms., construct a line of length xyz cms. 15. Given a triangle ABC, construct a point P on BC such that the lengths of the perpendiculars from P to AB and AC are in the ratio 2:3. 16. ABC is an equilateral triangle of side 5 cms., construct a point P inside it such that the perpendiculars from P to BC, CA, AB are in the ratio 1:2:3. Measure AP. 17. Draw any triangle ABC, use the method indicated in Sig. 210 to construct a triangle XYZ similar to triangle ABC and such that XY=2AB. A _- PROPORTION AND SIMILAR FIGURES 177 18. Given a quadrilateral ABCD, construct a similar quadrilateral each side of which is of the corresponding side of ABCD. 19. Given a triangle ABC and its median AD, construct a similar triangle XYZ and its median XW, such that XW=$AD. 20. Construct an equilateral triangle s?oh that the length )f ohe line joining one vertex to a point of trise^tiou of the opposite side is 2" ; measure its side. 21. Using a protractor, construct a regular jxmtagon such that the perpendicular from one corner to the opposite side is of length 7 cms. ; measure its side. 22. Construct a square ABCD, given that the length of the lino joining A to the mid-point of BC is 3"; measure its side. 23. Construct a triangle ABC, given ^BAC=54, /.ABC = 48, and the sum of the three medians is 12 cms. Measure AB. 24. Inscribe in a given triangle a triangle whose sides are parallel to the sides of another given triangle. 25. Given two radii OA, OB of a circle, centre O ; construct a square such that one vertex lies on OA, one vertex on OB, and the remaining vertices on the arc AB. 26. Inscribe a regular octagon in a square. 27. Inscribe in a given triangle ABC an equilateral triangle, one side of which is perpendicular to BC. 28. Construct a circle to touch two given lines and a given circle, centre O, radius a. [Draw two lines parallel to the given lines at a distance a from them : construct a circle to touch these lines and pass through O. Its centre is the centre of the required circle.] 29. Draw a line AB and take a point O V from it; P is a variable point on AB; Q is a point such that OQ = OP and Z.POQ = 50. Construct the locus of Q. [The locus of Q is obtained by revolving AB about O through 50.] 30. ABC is a given triangle ; P is a variable point on BC ; Q is a point such that the triangles ABC, APQ are similar. Construct the locus of Q. [Use the idea of ex. 29.] 31. APQ is a triangle of given shape ; A is a fixed point, P moves on a fixed circle ; construct the locus of Q. [Use the idea of ex. 29.] 12 CONCISE GEOMETRY 32*. Given a triangle ABC and a point D on BC, construct points P, Q on AB, AC such that DPQ is an equilateral triangle. 33*. ABC is a straight rod whose ends A, C move along two perpendicular lines OX, OY; AB=6 cms., BC = 3 cms, Draw the position of the rod when it maks an angle of 30 with OX, and construct the direction in which B is moving at this instant. 34*. AB and BC are two equal rods hinged together at B ; the end A is fixed and C is made to move along a fixed line AX ; D is the mid-point of the rod BC ; construct the direction in which D is moving when /. BAG = 45. 35*. A piece of cardboard in the shape of a triangle ABC moves so that AB and AC always touch two given fixed pins E, F ; draw the triangle in any position and construct the direction in which A is moving at that instant. THE MEAN PROPORTIONAL EXERCISE XXXVI L Construct a mean proportional between 5 andjS ; measure it. 2. Construct a line of length ^/43 cms. [Don't take a mean between 1 and 43, this leads to inaccurate drawing ; take numbers closer together, such as 5 and 8*6, 4 /- = &'($.] 3. Find graphically ^37. 4. Solve graphically the equation (x - 3) 2 = 19. 5. Draw a rectangle of sides 4 cms., 7 cms., and construct a square of equal area ; measure its^side. 6. Construct a square equal in area to an equilateral triangle of side 5 cms. ; measure its side. 7. Construct a square equal in area to a quadrilateral ABCD given AB = BC = 4, CD = 6, DA = 7, AC = 6 cms. ; measure its side. 8. Draw a line AB ; construct a point P on AB such that AP = 1 AB J% 9. Draw a circle, centre O ; construct a concentric circle whose area is one-third of the first circle. THE MEAN PROPORTIONAL 179 10. Given a triangle ABC, construct a line parallel to BO, cutting AB, AC at P, Q so that AAPQ = JAABC. 11. Given a quadrilateral ABCD, construct a similar quadrilateral with its area f- of the area of AC CD. 12. Given a triangle ABC, construct ai< rqv'uateral triangle of equal area. 13. Given three linos whose lengths are a, 6, c cms., construct a x 6 2 line of length x cms. such that - ~ ^. a c 2 14. Given two equilateral triangles, construct an equilateral triangle whose area is the sum of their areas. 15. Construct a circle to pass through two given points A, B and touch a given line CD. Use the method indicated in Fig. 211 and obtain two solutions. C ~' / P D FIG. 211. 16. Given a circle and two points A, B outside it, construct a point P on AB such that PA . PB is equal to the square of the tangent from P to the circle. 17. Construct a circle to pass through two given points A, B and to touch a given circle. 18. Given four points A, B, C, D in order on a straight line, construct a point P^on BC such that PA . PB = PC . PD. 19. Solve graphically the equations x y = 5, xy= 16. 20. OA, OB are two lines such that OA = 6 cms., L AOB = 40 ; construct a circle touching OA at A and intercepting on OB a length of 5 cms. 21. Construct a circle to pass through a given point, touch a given circle and have its centre on a given line. 22. Given three circles, each external to the others, construct a 180 CONCISE GEOMETRY point such that the tangents from it to the three circles are of equal length. 23. Draw a circle of radius 5 cms. and take a point A 3 cms. from the centre ; construct a chord PQ of the circle passing through A such that PA = |AQ. MISCELLANEOUS CONSTRUCTIONS IV EXERCISE XXXVII 1. Draw a line AB; if AB is of length x inches, construct a line of length x 2 inches. 2. AB, CD are two given parallel lines, and O is any given point ; construct a line OPQ, cutting AB, CD at P, Q so that AP : CQ is equal t;> a given ratio. 3. ABC is a given equilateral triangle of side 5 cms. ; construct a line outside it such that the perpendiculars from A, B, C to the line are in the ratio 2:3:4 and measure the last. 4. Construct a triangle ABC, given /. BAG = 48, z_BCA = 73, and the median BE = 5 cms. ; measure AC. 5. Construct a triangle ABC, given L ABC =62, ^ACB = 75 , and AB BC = 2 cms. ; measure BC. 6. Inscribe in a given triangle a rectangle having one side double the other. 7. Draw a triangle of sides 5, 6, 7 cms. and construct a square of equal area; measure its side. Check your result from the formula / s (s a) (s - b) (* - c). 8. Divide a square into three parts of equal area by lines parallel to one diagonal. 9. Given a triangle ABC, construct a line parallel to the bisector of L BAC and bisecting the area of AABC. 10. Given two lines AB and CD, construct a point P on AB produced such that PA . PB = CD 2 . REVISION PAPERS BOOK I 1. It requires four complete turns of the handle to wind up a bucket from the bottom of a well 24 feet deep. Through what angle must the handle be turned to raise the bucket 5 feet. 2. The angles of a triangle are in the ratio 1:3:5. Find them. 3. ACB is a straight line ; ABX, ACY are equilateral triangles on opposite sides of AB ; prove CX = BY. 4. ABCD is a quadrilateral ; ADCX, BODY are parallelograms ; prove that XY bisects AB. II 5. If the reflex angle AOB is four times the acute angle AOB, find ZAOB. 6. In A ABC, L BAG = 44, L ABC =112; find the angle be- tween the lines which bisect /.ABC and /.ACB. 7. The base BC of an isosceles triangle ABC is produced to D so that CD = CA, prove Z.ABD = 2Z.ADB. 8. ABCD is a parallelogram ; P is the mid-point of AB ; CP and DA are produced to meet at Q ; DP and CB are produced to meet at R ; prove QR = CD. Ill 9. 2LAOB = #; AO is produced to C; OP bisects Z.BOC; OQ bisects /. AOB ; calculate reflex angle POQ. 10. In AABC, L ABC = 35, Z.ACB = 75; the perpendiculars from B, C to AC, AB cut at O. Find 21 BOC. 181 182 CONCISE GEOMETRY 11. The bisector of the angle BAG cuts BC at D ; through C a line is drawn parallel to DA to meet BA produced at P ; prove AP = AC. 12. ABC is an acute-angled triangle; BANK, CAXY are squares outside the triangle ; prove that the acute angle between BH and CX equals 90 - Z. BAC. IV 13. Find the sum of the interior angles of a 15-sided convex polygon. 14. The sum of one pair of angles of a triangle is 100, and the difference of another pair is 60 ; prove that the triangle is isosceles. 15. ABC is a triangle right-angled at C ; P is a point on AB such that Z.PCB=Z.PBC; prove Z.PCA= |Z_BPC. 16. O is a point inside an equilateral triangle ABC; CAP is an equilateral triangle such that O and P are on opposite sides of AB; prove BP = OC. 17. If a ship travels due east or west one sea mile, her longitude alters 1 minute if on the equator, and 2 minutes if in latitude 60. Find her longitude if she starts (i) at lat. 0, long. 2 E. and steams 200 miles west ; (ii) at iat. 60 N., long. 2 W. and steams 150 miles east. 18. The bisectors of Z.s ABC, ACB of A ABC meet at O; if L BOC =135, prove /. BAC = 90. 19. In AABC, Z.ACB = 3/.ABC ; from AB a pait AD is cut off equal to AC ; prove CD = DB. 20. In AABC, AB = AC ; from any point P on AB a line is drawn perpendicular to BC and meets CA produced in Q ; prove AP = AQ. VI 21. O is a point outside a line ABCD such that OA = AB, OB = BC, OC = CD; Z.BOC = a; calculate Z.OAD and Z.ODA in terms of x. KEVISION PAPERS 1P3 22. In Fig. 144, page 137, if OQ bisects Z.AOC, pvo\a </ BOC 23. ABCD is a quadrilateral* DA=-DB DC; prov* / BAC-f 24. ABCD is a parallelogram; BP, DQ an two parallel lines cutting AC at P, Q ; prove BQ is p railel to DP. VII 25. In AABC, Z.BAC=115, Z.BCA=20; AD is the per- pendicular from A to BC ; prove AD = DB. 26. In Fig. 212, AB is parallel to ED; prove that reflex Z.EDC -reflex Z.ABC= /.BCD. 27. ABCD is a quadrilateral ; Z. ABC = Z. ADC = 90 ; prove that the bisectors of /.a DAB, DCB are parallel. 28. In AABC, Z.ABC = 90, Z.ACB = 60; prove AC=2BC. VIII 29. Two equilateral triangles ABC, AYZ lie outside each other ; if Z.CAY= 15, find the angle at which YZ cuts BC. 30. In AABC, AB = AC ; D is a point on AC such that BD = BC ; prove /.DBC= Z.BAC. 31. The altitudes BD, CE of AABC meet at H ; if HB=HC, prove AB = AC. 184 CONCISE GEOMETRY 32. P, Q, R, S are points on the sides AB, BC, CD, DA of a square ; if PR is perpendicular to QS, prove PR = QS. IX 33. In Fig. 213, express sr in terms of a, />, c. FIG. 213. 34. D is any point on the bisector of /.BAG; DP, DQ are drawn parallel to AB, AC to meet AC, AB at P, Q ; prove DP = DQ. 35. ABC is a A ; D, E are points on BC such that /. BAD = Z.CAE; if AD = AE, prove AB = AC. 36. ABCD is a square; the bisector of Z.BCA cuts AB at P; PQ is the i>erpendicular from P to AC ; prove AQ= PB. X* 37. ABCDEFGH is a regular octagon ; calculate the angle at which AD cuts BF. 38. In A ABC, AD is perpendicular to BC and AP bisects ZLBAC ; if /.ABOZ.ACB, prove ZABC- L ACB = 2 /. PAD. 39. ABCD is a straight line such that AB=BC=CD; BPQC is a parallelogram; if BP=2BC, prove PD is perpendicular to AQ. 40. The sides AB, AC of AABC^are produced to D, E; AH, AK are lines parallel to the bisectors of /.s BCE, CBD meeting BC in H, K: prove AB-f AC = BC + HK. REVISION PAPERS XI* 41. In Fig. 214, express z in oerm. j of ^, 7>, x, y. FIG. 214. 42. AB, BC, CD, DE are successive sides of a regular w-sided polygon ; find the angle between AB and DE. 43. In A ABC, AB = AC ; BA is produced to E ; the bisector of Z.ACB meets AB at D; prove ^CDE = f Z.CAE. 44. In A ABC, Z.BAC = 90 ; O is the centre of the square BPQC external to the triangle ; prove that AC bisects /. BAC. XII* 45. B is 4 miles duo east of A ; a ship sailing from A to B against the wind takes the zigzag course shown in Fig. 215, her directions being alternately N. 30 E. and S. 30 E. ; what is the total distance she travels ? Fio. 215. 46. ABC is a triangular sheet of paper, Z.ABC ==40, Z.ACB = 75 ; the sheet is folded so that B coincides with C ; find the angle which the two" parts of AB make with each other in the folded position. 186 CONCISE GEOMETRY 47. In A ABC, AB = AC; the bisector of Z ABC meets AC at D; P is a point on AC produced so that /.ABP= Z.ADB; prove BC = CP. 48. ABC is a A ', BDEC is a square outside AABC ; lines through B, C parallel to AD, AE meet at P ; prove PA is perpendicular to BC. BOOKS I, II XIII 49. AD, BE are altitudes of AABC; BC = 5 cms., CA=6 cms., AD = 4*5 cms. ; find BE. 50. ABC is an equilateral triangle ; P, Q are points on BC, CA such thatBP-CQ; AP cuts BQ at R; prove ,/ ARB-120 . 51. P is a variable point on a circle, centre O, radius a; C is a fixed point at a distance b from O ; find the greatest and least possible lengths of CP. 52. ABCD is a quadrilateral ; if AACD = ABCD, prove AABC = AABD. 53. Find in terms of x, XIV the area of Fig. 216. 90 90 y FIG. 216. 54. In AABC, AB = AC ; a line PQR cuts AC produced, AB, BC at R, P, Q ; if PQ = QR, prove AP-f AR==2AC. 55. The diagonals of the quadrilateral ABCD cut at O; if AAOD = ABOC, prove AS AOB, COD are equiangular. 56. In AABC, Z.BAC = 90, AB = 5 cms., AC = 8 cms.; find the area of the triangle and the length of its altitude AD. REVISION PAPERS 18? XV 57. Find in sq. cms. the area, making any poiihtr'u,ti m and measurements, of Fig. 217. FIG. 217. 58. ABODE is a regular pentagon; BD cuts CE at P; prove BP=BA. 59. The hypotenuse of a right-angled triangle is ( x* + -~) inches v #27 long, and one of the other sides is ( x 2 - \ inches. Find \ X 2/ the third side. 60. The side BC of the parallelogram ABCD is produced to any point K; prove AABK = quad. ACKD. XVI 61. ABCD is a parallelogram of area 24 sq. cms. ; its diagonals intersect at O; AB = 4*5 cms.; find the distance of O from CD. 62. In A ABC, Z. BAG = 90; BDEC is a square outside A ABC; DX is the perpendicular from D to AC ; prove DX = AB 4- AC. 63. BE, CF are altitudes of A ABC; prove = . AC CF 64. AD is an altitude of A ABC; AB = 7, AC = 5, BC = 8; if BD = #, DC = y, prove a; 2 y 2 =24, and find , y ; find also the area of A ABC. 188 CONCISE GEOMETRY XVII 65. In Fig. 218, ABCD is a quadrilateral inscribed in a rectangle; find the area of ABCD in terms of p, q, r, s 9 x, y. 66. In A ABC, BAG =90; P, Q are points on BC such that CA = CP and BA = BQ ; prove L PAQ = 45. 67. ABCD is a quadrilateral; /. ABC = L ADC = 90 ; AP, AQ are drawn parallel to CD, CB, cutting CB, CD at P, Q ; prove QA . AB = PA . AD. [Use area formulae.] 68. What is the length of the diagonal of a box whose sides are , 4", 12" XVIII 69. AD, BE, CF are the altitudes of A ABC; AB = 5# cms., BC = BOJ cms., CA = 3x cms., AD = 7 '5 cms. ; find BE, CF. 70. The base BC of the triangle ABC is produced to D ; the lines bisecting /.s ABD, ACD meet at P; a line through P parallel to BC cuts AB, AC at Q, R ; prove QR= BQ~CR. 71. ABCD is a rhombus; P, Q are points on BC, CD such that BP = CQ; AP cuts BQ at O; prove AAOB = quad. OPCQ. 72. In Fig. 219, AB=2", BC = 4", CD=1"; if PD 2 =2PA 2 , find PB. 90 Fio. 219. REVISION PAPERS 189 XIX 73. Soundings are taken at inte*-* ri v)s of 4 feet acio. * a river 40 feet wide, starting 4 feet from on 1 1* *,nk, and the following depths in feet are obtained in oij*-tf 60, 9'3, 9'9, 8'2, 8*4, 10-2, 10*5, 7*8, 4*5; find approximately the area of the river's cross-section. 74. In the A ABC, AB-BC and < ABC = 90; the bisector of Z. BAC cuts BC at D ; prove AB + BD = AC. 75. ABCD is a parallelogram; P is the mid-point of AD; AB is produced to Q so that AB = BQ ; prove ABCD = 2 APQD. 76. In A ABC, Z. BAC = 90; P is the mid-point of AC; PN is drawn perpendicular to BC; prove BN 2 =BA 2 + CN 2 . XX 77. ABCD is a parallelogram; AB-4 cms., BC = 5aj cms.; the distance of A from BC is 6 cms. ; find the distance of D from AB. 78. In Fig. 220, AB = BP = 4", BC=PQ = 3", AC=BQ = 5"; calculate the area common to AS ABC, BPQ. 79. In A ABC, AB = AC; P is any point on BC; Q, R are the mid-points of BP, PC ; QX, RY are drawn perpendicular to BC and cut AB, AC at X, Y ; prove BX = AY. 80. ABC is an equilateral triangle ; BC is bisected at D and pro- duced to E so that CE = CD, prove AE 2 =7EC 2 . 190 CONCISE GEOMETRY XXI 81. In Fig. 221, the triangle ABC is inscribed in a rectangle: find its area and the distance of A from the mid-point of BC. FIQ. 221. 82. A, B are fixed points ; X is a variable point such that /. AXB is obtuse ; the perpendicular bisectors of AX, BX cut AB at Y, Z ; prove that the perimeter of AXYZ is constant. 83. ABC is a A ; a line XY parallel to BC cuts AB, AC at X, Y and is produced to Z so that XZ=BC; prove ABXY = AAYZ. 84. The sides of a triangle are 8 cms., 9 cms., 12 cms. Is it obtuse-angled *? XXII* 85. ABC is a triangle of area 24 sq. cms.; AB = 8 cms., AC = 9 cms.; D is a point on BC such that BD=BC; find the distance of D from AB. 86. O is a point inside A ABC such that OA = AC, prove that BA>AC. 87. ABCD is a quadrilateral; AB is parallel to CD; BP, CP are drawn parallel to AC, AD to meet at P; prove APDC = ABD. 88. The length, breadth, and height of a room are each 10 feet; CAE, DBF are two vertical lines bisecting opposite walls, C, D being on the ceiling and E, F on the floor ; CA = x feet, DB = 4 feet. Find in terms of x the shortest path from A to B (i) along these two walls and the ceiling ; (ii) along these two walls and one other wall. What is the condition that route (ii) is shorter than route (i) ? REVISION PAPERS 91 XX III 89. In Fig. 222, AB---9", BC-8 A , CD -7"; if AP PD, calcu- late BP. oo B P FIG. 222. 90. ABC is a A ; AP is the perpendicular from A to the^bisectcr of L ABC; PQ is drawn parallel to BC to cut AB at Q; prove AQ = QB = PQ. 91. ABP, ABQ are equivalent triangles on opposite sides of AB ; PR is drawn parallel to BQ to meet AB at R ; prove QR is parallel to PB. 92. In A ABC, Z.BAC-90 ; H, K are the mid-points of A B, AC ; prove that BH 2 + HK 2 + KC 2 = BC 2 . XXIV* 93. The angles at the corners of Fig. 223 are all right angles. Construct a line parallel to AB to bisect the given figure. [The fact in Ex. XXXI, No. 22, may be useful.] FiO. 223. 94. In A ABC, L BAG = 90; P, Q are the centres of the two squares which can be described on BC; prove that the distances of P, Q from AB are J(AB+AC). 192 CONCISE GEOMETRY 95. ABCD is a parallelogram ; any line parallel to BA cuts BC, AC, AD at X, Y, Z ; prove AAXY = ADYZ. 96. In A ABC, ZACB = 90; AD is a median; prove that AB 2 3BD 2 . BOOKS I-III XXV 97. The side BC of an equilateral triangle ABC is produced to D so that CD = SBC ; prove AD 2 = 13AB 2 . 98. ABCD is a quadrilateral; if Z.ABC+ L ADC -180, prove that the perpendicular bisectors of AC, BD, AB are con- current. 99. ABCD is a quadrilateral inscribed in a circle ; AC is a diameter; /.BAG = 43; find ZADB. 100. Two circles ABPQ, ABR intersect at A, B ; BP is a tangent to circle ABR; RAQ is a straight line; prove PQ is parallel to BR. XXVI 101. ABC is a A ; H, K are the mid-points of AB, AC; P, Q are points on BC such that BP = ^BC = jBQ; prove PH = QK. 102. Find the remaining angles in Fig. 224. FIG. 224. 103. ABCD is a parallelogram ; the circle through A, B, C cuts CD at P; prove AP = AD. 104. APB, AQB are two circles; AP is a tangent to circle AQB; PBQ is a straight line; prove that AQ is parallel to the tangent at P. REVISION PAPERS XXVII 105. ABCD is a square ; P is a point on AB such that AP = JAB ; Q is a point on PC such tbu PQ = ^PC; pro e APQD = $ABCD. 106. AOB is a diameter of a circle perpeii'iii ular to a chord POQ ; AO = &, PQ = et; tind AB in terms of a, h. 107. The side AB of a cyclic quadrilateral ABCD is produced to E; Z.DBE-140 , Z.AOC=iOO, Z.ACB = 45; find Z.BAC, /iCAD. 108. In A ABC, /.BAG = 90 ; the circle on AB as diameter cuts BC at D ; the tangent at D cuts AC at P ; prove PD = PC. XXVIII 109. In quadrilateral ABCD, AB = 7", CD -II", Z.BAD= Z.ADC -90, /.BCD = 60; calculate AC. 110. Two chords AB, DC of a circle, centre O, are produced to meet at E; Z.CBE-75 , Z.CEB = 22, Z.AOD=144; prove /. AOB = L BAC. 111. In Fig. 225, O is the centre and TQ bisects Z.OTP; prove 112. PAB, PBC, PCA are three unequal circles; from any point D on the circle PBC, lines DB, DC are drawn and produced to meet the circles PBA, PCA again at X, Y ; prove XAY is a straight line. 13 194 CONCISE GEOMETRY XXIX 113. In AABC, ZACB = 90, AC = 2CB; CD is an altitude; prove by using the figure of Pythagoras' theorem or other- wise that AD = 4DB. 114. In Fig. 226, O is the centre of the circle; PQ and PT are equally inclined to TO ; prove Z.QOT= 3 Z. POT. FIG. 226. 115. AOB is a chord of a circle ABC ; T is a point on the tangent at A; the tangent at B meets TO produced at P; /.ATO = 35, Z.BOT=115; find Z.BPT. 116. In AABC, AB = AC ; the circle on AB as diameter cuts BC at P; prove. BP= PC. XXX 117. X, Y, Z are any points on the sides BC, CA, AB of the triangle ABC ; prove that AX + BY + CZ > J(BC -f CA -f AB). 118. A, B, C, D are the first milestones on four straight roads running from a town X ; A is due north of D and north- west of B. C is E. 20 S. of D ; find the bearing of B from C. 119. ABCD is a quadrilateral inscribed in a circle, centre O; if AC bisects /. BAD, prove that OC is perpendicular to BD. 120. A diameter AB of a circle APB is produced to any point T; TP is a tangent ; prove L BTP + 2 L BPT = 90. XXXI 121. ABCD is a rectangle; P is any point on CD; prove that quad. ABCP - A APD =* AD . CP. 122. ABCD is a circle; if arc ABC = arc ADC, find /.ADC. REVISION PAPERS 123. A, B, C are points on a circle, centre O; BO, CO are pro duced to meet AC, AB at P, Q; prove Z.BPC + /.BQC = 3 L BAC. 124. In Fig. 227, AB is a diameter; prove AH = BK. P Fro. 227. XXXII 125. In A ABC, Z BAC = 90; AD is an altitude; prove that JL-J-+..L AD 2 AB 2 AC 2 ' 126. ABCD is a square inscribed in a circle; P is any point on the minor arc AB ; prove /. APB = 3 /. BPC. 127. ABC is a triangle inscribed in a circle ; the bisector of /. BAC meets the circle at P ; I is a point on PA between P and A such that PI = PB ; prove L IBA = L IBC. 128. Two circles, centres A, B, cut at X, Y ; XP, XQ are the tangents at X ; prove L AXB is equal or supplementary to XXXIII 129. ABCD is a parallelogram; P is any point on CD ; PA, PB, CB, AD cut any line parallel to AB at X, Y, Z, W ; prove DCZW=2AAPY. 130. In Fig. 228, O is the centre, PQ = AO, Z.AOQ = 90; prove arc BR = 3 arc AP. J* Fro. 228. 196 CONCISE GEOMETRY 131. A rectangular strip of cardboard is 7 inches wide, 4 feet long; how many circular discs each of radius 2 inches can be cut out of it 1 132. AB, CD are parallel chords of a circle ABDC, centre O ; prove Z.AOC equals angle between AD and BC. XXXIV 133. Two metre rules AOB, COD cross one another at right angles : the zero graduations are at A, C ; a straight edge XY, half a metre long, moves with one end X on OB and the other end Y on OD ; when the readings for X are 50, 40 cms., those for Y are 50, 60 cms. respectively. Find the readings at O. 134. Two circles PARB, QASB intersect at A, B; a line PQRS cuts one at P, R and the other at Q, S; prove Z.PAQ = Z.RBS. 135. In A ABC, Z.BAC = 90; D is the mid-point of BC ; a circle touches BC at D, passes through A and cuts AC again at E ; prove arc AD = 2 arc DE. 136. Two circular cylinders of radii 2", 6" are bound tightly together with their axes parallel by an elastic band. Find its stretched length. XXXV 137. In Fig. 229, BC is an arc of radius 8" whose centre lies on OB produced; 08 = 9", L AOB = 90; calculate the radius of a circle touching AO, OB and arc BC. O B FIG. 229. 138. ABCD is a parallelogram ; AB, CB are produced to X, Y ; P is any point within the angle XBY ; prove A PCD - APAB = AABC. 139. A 1 A 2 A 3 . . . A^ is a refeuar polygon of 20 sides, prove that A! A 8 is perpendicular to A 3 A 16 . 'REVISION PAPERS 197 140. A, B, C are three points on a circle ; the tangent at A BC produced at D; prove that the bisect w of ^b BM3, BDA are at right angles. XXXVI HI. In AABC, /. ABC -90, ^BAC = I/*, Jie bisector of L ACB meets AB at P ; prov AP 2 = 2PB 2 . 142. The diameter AB of a circle is i induced to any point P; a line is drawn from P touching Uio circle at Q and cutting the tangent at A in R; prove Z.BQP = J/.ARP. 143. In AABC, AB = AC and Z. BAG is obtuse ; a circle is drawn touching AC at A, passing through B and cutting BC again at P ; prove arc AB = 2 arc AP. 144. The volume of a circular cylinder is V cub. in. and the area of its curved surface is S sq. in. ; find its radius in terms of V, S. BOOKS I-IV XXXVII 145. In Fig. 230, if ^ADC= Z.BEA- Z.CFB, prove that the triangles ABC, XYZ are equiangular. B DC FIG. 230. 146. The tangent at a point R of a circle meets a chord PQ at T; O is the centre; E is the mid-point of PQ; prove /.ROT = RET. 147. A line AB, 8 cms. long, is clivided internally and externally in the ratio 3 : 1 at P, Q respectively ; find PQ : AB. 198 CONCISE GEOMETRY 148. ABCD is a quadrilateral; a line AF parallel to BC meets BD at F ; a line BE parallel to AD meets AC at E ; prove EF is parallel to CD. XXXVIII 149. The sides AB, BC, CA of A ABC are produced their own lengths to X, Y, Z; prove AXYZ = 7AABC. 150. ABCD is a quadrilateral ; the circles on AB, BC as diameters intersect again at P; the circles on AD, DC as diameters intersect again at Q ; prove BP is parallel to DQ. 151. A town occupies an oval area of length 2400 yards, breadth 1000 yards : a plan is made of it on a rectangular sheet of paper 18" Jong, 12" wide. What is the best scale to choose? 152. ABC is a triangle inscribed in a circle; AD is an altitude; AP is a diameter : prove - = and complete the equation ' l AP AC AB = AP' XXXIX 153. AB is a diameter of a circle; AOC, BOE are two chords such that L CAB = Z. EBA = 22 ; prove that AC 2 = 2OC 2 . 154. PQ is a chord of a circle ; T is a point on the tangent at P such that PT= PQ ; TQ cuts the circle at R ; prove L RPT = 155. In Fig. 231, AB, CD, EF are parallel; AD=7 77 , DF = CE - 4" ; find BC. If EF - 2", AB = 3", find CD. 156. AB, DC are parallel sides of the trapezium ABCD; AC cuts DB at O ; the line through O parallel to AB cuts AD, BC at P, Q ; prove PO - OQ. REVISION PAPERS XL 157. In A ABC, AB-AC arid L BAG =120; the perpendicular bisector of AB cuts BC at X; prove BC = 3BX, 158. AOB, COD are two perj>endicuu< cnorcLo/ a rir le; prove that arc AC -f arc BD equals h'tlf the ir ip'itereiice. 159. A light is placed 4' in front oi a cirtii'ai hoi 3" in diameter in a partition; find the diameter of the illuminated part of a wall 5' behind tho partition pnd parallel to it. 160. ABC is a triangle inscribed in a circle; AB = AC; AP is a chord cutting BC at Q ; prove AP . AQ --- AB' 2 . XLI 161. In AABC, ZBAC=90, Z. ABC- 45; AB is produced to D so that AD . DB = AB 2 ; prove that the perpendicular bisector of CD bisects AB. 162. ABCD ia acyclic quadrilateral; AC cuts BD at O; it CD touches the circle OAD, prove that CB touches the circle OAB. 163. ABCDEF is a straight line; AB : BC : CD : DE : EF = 2:3:7:4:5; find the ratios AD and ~. DF AF 164. ABCD is a parallelogram; a line through A cuts BD, BC, CD at E, F, G; prove *p = ^. XLII 165. AB is a diameter of a circle APB; the tangent at A meets BP at Q ; prove that the tangent at P bisects AQ. 166. PAQ, PBQ, PCQ are three equal angles on the same side of PQ ; the bisectors of L s PAQ, PBQ meet at H ; prove that CH bisects L PCQ. 167. Two triangles are equiangular: the sides of one are 3 cms., 5 cms., 7 cms. ; the perimeter of the other is 2J feet; find its sides. 168. Two lines OAB, OCD cut a circle at A, B, C, D; H, K are points on OB, OD such lhat OH = OC, OK=*OA; prove that HK is parallel to BD. 200 CONCISE GEOMETRY XLiri 169. C is the mid-point of AB ; P is any point on CB ; prove that AP 2 -PB 2 =2AB.CP. 170. A circular cylinder of height 6" is cut from a sphere of radius 4" ; find its greatest volume. 171. Show that the triangle whose vertices are (2, 1), (5, 1), (4, 2) is similar to the triangle whose vertices are (1, 1) (7, 1) (5, 3). 172. Two circles intersect at A, B ; the tangents at A meet the BC BA circles at C, D : prove = - - . BA BD XLIV 173. ABCD is a quadrilateral; AP is drawn equal and parallel to BD; prove AAPC = quad. ABCD. 174. A circular cone is made from a sector of a circle of radius 6" and angle 240 ; find its height. 175. A straight rod AB, 3' 9" long, is fixed under water with A 2' 6" and B 9" below the surface ; what is the depth of a point C on the rod where AC = I' 1 176. ABCD is a straight line; O is a point outside it; a line through B parallel to OD cuts OA, OC at P, Q ; if PB= BQ, AB _ AD BC~~CD" XLV 177. In Fig. 232, OA, AB are two rods hinged together at A; the end O is fixed, and AO can turn freely about it ; the end B is constrained to slide in a fixed groove OC. OA = 3', AB = 4' ; find the greatest length of the groove which B can travel over, and calculate the distance of B from O when AB makes the largest possible angle with OC. C B Fir,. 232. REVISION PAPERS 20 i 178. ABC is a triangle inscribed in a circle; P, Q, R are the mid- points of the arcs BC, CA, AB ; prove AP is perpendicular toQR. 179. AOXB, COYD are two straight lines; AC, XY, BD are parallel lines cutting them j AX = 7, XB = 3, AC = 2, BD = 4 ; find XY. 180. P is any point on the common chord of two circles, centres A, B; HPK and XPY are chords of the two circles per- pendicular to PA, PB respectively; prove HK = XY. XLVI 181. ABC is a triangle inscribed in a circle ; the internal and external bisectors of /. BAG cut BC at P, Q ; prove that the tangent at A bisects PQ. 182. A circle of radius 4 cms. touches two perpendicular lines; calculate the radius of the circle touching this circle and the two lines. 183. ABCD is a rectangle; AB = 8", BC=5"; P is a point inside it whose distances from AD, AB arc 2", I" ; DP is produced to meet AB at E ; CE cuts AD at F ; calculate EB, AF. 184. Two lines CAB, OCD meet a circle at A, B, C, D; prove OA . CD AD 2 that OB . OC BC 2 XLVil* 185. ABC is an equilateral triangle ; P is any point on BC; AC is produced to Q so that CQ = BP ; prove AP = PQ. 186. AB is a diameter of a circle APB; AH, BK are the per- pendiculars from A, B to the tangent at P ; prove that AH 4- BK = AB. 187. A chord AB of a circle ABT is produced to O; OT is a tangent; OA=6", OT = 4", AT- 3", find BT. 188. AB, DC are parallel sides of the trapezium ABCD ; AC cuts BD at E ; DA, CB are produced to meet at F ; EF cuts AB, DC at P, Q ; prove SL. == SL. 202 CONCISE GEOMETRY XLVIII* 189. A brick rusts on the ground and an equal brick is propped up against it as in Fig. 233. The bricks are V by 2". Calculate the height of each corner of the second brick above the ground, if AB = 1 J". 190. Prove that the area of a square inscribed in a given semi- circle is | of the area, of the square inscribed in the whole circle. 191. The bisector of Z.BAC cuts BC at D; the line through D BY BD perpendicular to DA cuts AB, AC at Y, Z ; prove = - . CZ DC 192. A chord AD is parallel to a diameter BC of a circle; the tangent at C meets AD at E ; prove BC . AE = BD 2 . XLIX* 193. A is a fixed point on a given circle ; a variable chord AP is produced to Q so that PQ is of constant length ; QR is drawn perpendicular to AQ ; prove that QR touches a fixed circle. 194. Four equal circular cylinders, diameter 4*, length 5", are packed in a rectangular box ; what is the least amount of unoccupied space in the box 1 195. A rectangular sheet of paper ABCD is folded so that B falls on CD and the crease passes through A ; AB = 10", BC = 6" ; find the distance of the new position of B from C. If the crease meets BC at Q, find CQ. 196. ABCD is a parallelogram ; a line through A cuts BD, CD, BC PO PD 2 * in P, Q, R; prove -| = . REVISION PAPERS 203 197. In Fig. 234, ABCD is a rectangle ; BP= 2CQ; AD = 2AB = 6". The area of APQD is 10 sq. in. ; find BP. FIG. '234. 198. ABC is a triangle inscribed in a circle ; the tangents at B, C meet at T ; a line through T parallel to the tangent at A meets AB, AC produced at D, E ; prove DT = TE. 199. A line HK parallel to BC cuts AB, AC at H, K ; the distance between HK and BC is 5 cms. ; the areas of AHK and HKCB are 9 sq. cms., 40 sq. cms. ; find HK. 200. In A ABC, I is the in-centre and l a is the ex-centre corre- sponding to BC ; prove Al . Al x = AB . AC. 204 CONCISE GEOMETRY WHEN learning propositions, do not use the figure printed in the book, but draw your own figure instead. It is more trouble but gives bettor results. For this reason, no attempt has been made to arrange the whole proof of every theorem on the same page as the corresponding figure. A freehand figure is good enough. PROOFS OF THEOREMS BOOK I DEFINITION. If C is any point on the straight line AB, and if a line CD is drawn so that the angles ACD, BCD are equal, each is called a right angle. D AC B Fro. 235. Therefore if C is any point on the straight line AB, the angle ACB is equal to two right angles, or 180. THEOREM 1 (1) If one straight line stands on another straight line, the sum of the two adjacent angles is two right angles. (2) If at a point in a straight line, two other straight lines, on opposite sides of it, make the adjacent angles together equal to two right angles, these two straight lines are in the same straight line. E> FIG. 1. 205 206 CONCISE GEOMETRY (1) Given CE meets AB at C. To Prove L ACE + ^ BCE = 1 80. L ACE+ ^BCE^ L. ACB = 180, since ACB is a st. line. Q.E.D. Ey ACB F FIG. 236. (2) Given L. ACE + L BCE = 180. To Prove ACB is a straight line. Produce AC to F. .'. L. ACE + L FCE - 180, since ACF is a st. line. But L. ACE 4- L. BCE- 180, given. /. L. ACE -h L. FCE = L. ACE + - BCE. /. ^FCE= Z.BCE. CB falls along CF. But ACF is a st. line ; .*. ACB is a st. line. Q.E.D. THEOREM 2 If two straight lines intersect, the vertically opposite angles a equal. FIG. 2. To Prove that x = y and a = /3. x 4- a = 1 80 adjacent angles. a + y = 1 80 adjacent angles. Similarly a = j8. Q.K.D. For riders on Theorems 1-2, see page 2. PROOFS OF THEOREMS 207 THEOREM 3 If two triangles have two sides of one equal respectively to two sides of the other, and if the included angles are equal, then the triangles are congruent. A P B C Q R FIG. 2#. Given AB = PQ, AC - PR, L. BAG = L QPR. To Prove A ABC = A PQR. Apply the triangle ABC to the triangle PQR, so that A falls on P and the line AB along the line PQ ; Since AB = PQ, .'. B falls on Q. Also since AB falls along PQ and L BAG = L QPR, .". AC falls along PR. But AC = PR, /. C falls on R. .". the triangle ABC coincides with the triangle PQR. /. AABC-APQR. Q.K.D. For riders on Theorems 3, 9, 10, see page 16. THEOREM 4 If one side of a triangle is produced, the exterior angle is greater than either of the interior opposite angles. 208 CONCISE GEOMETRY BC is produced to D. To Prove L ACD > L ABC and L ACD > L BAG. Lot F be the middle point of AC. Join BF and produce it to G, so that BF= FG. Join CG. In the triangles AFB, CFG AF - FC and BF = FG, constr. L AFB =-- u CFG, vert. opp. .'. AAFB-ACFG. /. .LBAF- Z.GCF. But L DCA> its part L, GCF. .'. L DCA > L BAI; or L BAC. Similarly, if BC is bisected and if AC is produced to E, it can be proved that L BCE > L. ABC, But L ACD - L BCE, vert. opp. /. L ACD> . ABC. Q.E.D. DEFINITION. Straight lines which lie in t/ie same pl<me and which never meet, however far tlifj are produced either way, are called parallel straight lines. PLAYF AIR'S AXIOM. Through a given point, one and only one straight line can be drawn parallel to a given straight line. THEOREM 5 If one straight line cuts two other straight lines such that either (1) the alternate angles are equal, or (2) the corresponding angles are equal, or (3) the interior angles on the same side of the cutting line are supplementary, then the two straight lines are parallel. /A P Q PROOFS OF THEOREMS 209 ABCD cuts PQ, RS at B, C. (1) Given ^PBC= L BCS. To Prove PQ is parallel to RS. If PQ, RS are not parallel, they will meet when produced, at H, say. Since BCH is a triangle, ext. L. PBC > int. <L BCH, which is contrary to hypothesis. .". PQ cannot meet RS and is .*. parallel to it. Q.B.P. \?) Given L ABQ = L. BCS. To Prom PQ is parallel to RS. L. ABQ = L PBC, vert. opp. But L ABQ = L. BCS, given. /. -L PBC = L. BCS. /. by (1), PQ is parallel to RS. (3) Given L. QBC 4- SCB - 180. To Prove PQ is parallel to RS. L. QBC+ <- PBC = 180, adj. angles. But _ QBC + L. SCB - 180, given. .'. _ QBC + L. PBC = L QBC f L SCB. /. L. PBC - L. SCB. /. by (1), PQ is parallel to RS. Q.E.D. THEOREM 6 If a straight line cuts two parallel straight lines, Then (1) the alternate angles are equal; (2) the corresponding angles are equal ; (3) the interior angles on the same side of the cutting line are supplementary. V P B FIG. 239. 14 210 CONCISE GEOMETRY AB, CD are two parallel st. lines ; the line PS cuts them at Q, R. To Prove (1) Z.AQR- ^QRD. (2) L. PQB = L. QRD. (3) ^.BQR+ ^.QRD-180 . (1) If L AQR is not equal to L QRD, let the angle XQR be equal to L. QRD. But these are alternate angles. .*. QX is parallel to RD, .'. two intersecting lines QX, QA are both parallel to RD, which is impossible by PI ayf air's Axiom. L. AQR cannot be unequal to L QRD. (2) LPQB=- L AQR, vert. opp. But ^AQR= 1.QRD, alt. angles. (3) L. BQR + L AQR - 180, adj. angles. But L. AQR = L QRD, alt. angles. .'. ^BQR + L QRD = 180. Q.E.D. For riders on Theorems 5, 6, see page 6. \ THEOREM 7 ' (1) If a side of a triangle is produced, the exterior angle is equal to the sum of the two interior opposite angles. (2) The sum of the three angles of any triangle is two right angles. ABC is a triangle ; BC is produced to D. To Prove (1) L. ACD L CAB + L ABC. (2) L. CAB + L. ABC 4- L ACB = 180. PROOFS OF THEOREMS 211 (1) Let CF be drawn parallel to AB. L FCD = L ABC, corresp. angles* L ACF = L CAB, alt. angles, adding, L FCD + L ACF = L ABC + L CAB. .'. 1ACD= .LABC + /.CAB. (2) Add to each the angle ACB. .*. L ACD + ,1 ACB - L ABC + _ CAB + L. ACB. But _ ACD + L ACB = 180, adj. angles. .'. L ABC + 21 CAB + - ACB - 180. Q.K.D. THEOREM 8 (1) All the interior angles of a convex polygon, together with four right angles, are equal to twice as many right angles as the polygon has sides. (2) If all the sides of a convex polygon are produced in order, the sum of the exterior angles is four right angles. Let n be the number of sides of the polygon. (I) To Prove that the sum of the angles of the polygon + 4 rt. L. s = 2n rt. LS. Take any point O inside the polygon and join it to each vertex. The polygon is now divided into n triangles. But the sum of the angles of each triangle is 2 rt. L s. .". the sum of the angles of the n triangles is 2n rt. L s. But these angles make up 'all the angles of the polygon together with all the angles at O. 212 CONCISE GEOMETRY Now the sum of all the angles at O is 4 rt. L s. .". all the angles of the polygon -h 4 rt. <- s = 2n rt. L s. FIG. 242. (2) At each vertex, the interior _ + the exterior - - 2 rt. _ s. .". the sum of all the interior angles -j- the sum of all the exterior angles =- 2n rt. ^ s. ]}ut the HUIII of all the interior angles -f I rt. L. s = 2n rt. L s. .*. the sum of all the exterior *_ s = 4 rt. .L s. Q.E.D. 8(1) may also be vt--lod as follows : The sum of the interior angles of any convex polygon of n sides is 2w - 4 right angles. For riders on Theorems 7, 8, see page 10. TJIEOUKM 9 Two triangles are congruent if two angles and a side of one are respectively equal to two angles and the corresponding side of the other. A P Given either that BC^QR. PROOFS OF THEOREMS 213 or that FiC = QR. z_A3C= A. PQR. To Prove AABC== APQR. The sum of the three angles of any triangle is 180. .*. in each case, the remaining pair of angles is equal. Apply the triangle ABC to the triangle PQR so that B falls on Q and BC falls along QR. Since BC = QR, C falls on R. -A And since BC falls on QR aiul'-L ABC=- L. PQR, .". BA falls along QP. And since CB falls on RQ and L. ACB = L PRQ, /. CA falls along RP. /. A falls on P. .*. the triangle ABC coincides with the triangle PQR. .'. A ABC = A PQR. Q.E.D. THEOREM 10 (1) If two sides of a triangle are equal, then the angles opposite to those sides are equal. (2) If two angles of a triangle are equal, then the sides opposite to those angles are equal. A B D C FIG. 244. ABC is a triangle : let the line bisecting the angle BAC meet BC at D. (1) Given AB - AC. To Prwe L ACB = L ABC. In the As ABD, ACD. AB AC, given. AD is common. (2) 214 CONCISE GEOMETRY L BAD = L CAD, constr. .*. the AS are congruent. .'. L ABD - L ACD. Given L ABC = L ACB. To Prove AC = AB. In the AS ABD, ACD. i. ABD = L. ACD, given. L BAD = L CAD, constr. AD is common, the AS are congruent. .'. AB = AC. Q.E.D. For riders on Theorems 3, 9, 10 see page 15. THEOREM 11 Two triangles are congruent if the three sides of one are re- spectively equal to the three sides of the other. FIG. 245(2). Given that AB = XY, BC = YZ, CA = ZX. To Prove A ABC s? A XYZ. Place the triangle ABC so 'that B falls on Y and BC along YZ; /. since BC = YZ, C falls on Z. PROOFS OF THEOREMS 215 Let the point A fall at a point F on the opposite side of YZ to X. Join XF. Now YF = BA, constr. But BA^YX, given. .'. YF-YX But these are sides of the triangle YFX. .'. .YXF = /.YFX. Similarly, Z.ZXF- _ ZFX. .*. adding in Fig. 245(1) or subtracting in Fig. 245(2) ^_YXZ= <_YFZ. But L BAC= L YFZ, constr. .'. L. BAG - L YXZ. .'. in the AS ABC, XYZ AB = XY, given. AC = XZ, given. Z.BAC- ^.YXZ, proved. /. A ABC == A XYZ. J\ - - Q.B.D. THEOREM 12 Two right-angled triangles are congruent if the hypotenuse and side of one are respectively equal to the hypotenuse and a side of the other. A X c B t y F Fid. 246. Given /.ABC = 90= L. XYZ. AC = XZ. AB = XY. To Proe AABC== AXYZ. Place the triangle ABC so th^t A falls on X and AB falls along XY, and so thai C falls at some point F on the opposite side of XY to Z. 216 CONCISE GEOMETRY Since AB = XY, B falls on Y. L. XYF = L ABC = 90 and u XYZ = 90. .'. Z.XYF + ^ XYZ =180. ZYF is a straight line. But XF = AC, and AC is given equal to XZ. XZF is a triangle, in which XF XZ, But L XFY- L ACB, constr. .'. L. XZY - L. ACB. /. in the AS XYZ, ABC. L XYZ = L ABC, given. L. XZY = L. ACB, proved. XY - AB, given. /. A XYZ = A ABC. Q.E.D. THEOREM 13 (1) The opposite sides and angles of a parallelogram are equal. (2) Each diagonal bisects the parallelogram. B Fm. as Given ABCD is a parallelogram. To Prove (1) AB - CD and AD - BC. L. DAB = L DCB and L ABC = L. ADC. (2) AC and BD each bisect the parallelogram. Join BD. In the AS ADB, CBD L ADB = L CBD, alt. L s. L ABD = L. CDB, alt. _ s. BD is common. /. A ADB s A CBD. .'. AB = CD, AD = BC, c L DAB == L BCD and BD bisects the parallelogram. PROOFS OF THEOREMS 217 Similarly, by joining AC it may bo proved that L ABC = L ADC, and that AC bisects the parallelogram. Q.E.D. THEOREM 14 The diagonals of a parallelogram bisect one another. A B D C FIG. 29. The diagonals AC, BD of the parallelogram ABCD intersect at O. To Prove AC = OC and BO - CD. In the As AOD, COB, L DAO = L BCO, alt. L s. L ADO = L CBO, alt. L s. AD = BC, opp. sides of ||gram. /. A AOD = A COB. AO = CO and BO = DO. Q.E.D. THEOREM 15 The straight lines which join the ends of two equal and parallel straight lines towards the same parts are themselves equal and parallel. A B C D FIG. 30. Given AB is equal and parallel to CD. To Prove AC is equal and 1 parallel to CD Join BC. 218 CONCISE GEOMETRY In the As ABC, DOB AB = DC, given. BC is common. L ABC = L DCB alt. angles, AB being || to CD. .'. AABC-ADCB. .-. AC-DBand <L ACB = ^ DBC. J3ut these are alt. angles, .'. AC is parallel to DB. Q.E.D. This theorem can also be stated as follows : A quadrilateral which has one pair of equal and parallel sides is a parallelogram. Other tests for a parallelogram are : (1) If the diagonals of a quadrilateral bisect each other, it is a parallelogram. (2) If the opposite sides of a quadrilateral are equal, it is a parallelogram. (3) If the opposite angles of a quadrilateral are equal, it is a parallelogram. For riders on Theorems 11, 12, 13, 14, 15, see page 23. BOOK II THEOREM 16 (1) Parallelograms on the same base and between the same parallels are equal in area. (2) The area of a parallelogram is measured by the product of its base and its height. (1) Given ABCD, ABPQ are two 'parallelograms on the same base AB and between the same parallels AB, DP. PROOFS OF THEOREMS 219 To Prove that ABCD, ABPQ are equal in area. In the As AQD, BPC, L ADQ = L BCP, corresp. L. s ; AD, BC being || lines. L AQD= L BPC, corresp. L. s; AQ, BP being || lines. AD = BC, opp. sides Hgram. From the figure ABPD, subtract in succession each of the equal triangles BPC, AQD. .". the remaining figures ABCD, ABPQ are equal in area. D K C H Fia. 32. (2) If BH is the perpendicular from B to CD, the area of ABCD is measured by AB . BH. Complete the rectangle ABHK. The Hgram ABCD and the rectangle ABHK are on the same base and between the same parallels and are therefore equal in area. But the area of ABHK = AB . BH ; .'. the area of ABCD = AB . BH. Q.B.D. THEOREM 17 The area of a triangle is measured by half the product of the base and the height. A C FIG. 247. 220 CONCISE GEOMETRY Given that AD is the perpendicular from A to the base BC of the triangle ABC. To Prove that the area of A ABC = JAD . BC. Complete the parallelogram ABCK. Since the diagonal AC bisects the parallelogram ABCK, A ABC =* i parallelogram ABCK. But parallelogram ABCK = AD . BC ; /. AABC-IAD.BC. Q.E.D. THEOREM 18 (1) Triangles on the same base and between the same parallels are equal in area. (2) Triangles of equal area on the same base and on the same side of it are between the same parallels. (1) Given two triangles ABC, ABD on the same base AB and between the same parallels AB, CD. To Prove the triangles ABC, ABD are equal in area. Draw CH, DK perpendicular to AB or AB produced. But CH is parallel* to DK, since each is perpendicular to AB, and CD is given parallel to HK. CDKH is 9> parallelogram. CH = DK? opp. sides. A CAB equals A DAB in area. PROOFS OF THEOREMS 221 (2) Given two triangles ABC, ABD of equal area. To Prove CD is parallel to AB. Draw CH, DK perpendicular to AB or AB produced. Now ACAB = JCH . AB and A DAB = JDK . AB. .'. CH . AB = DK . AB. /. CH - DK. But CH is parallel to DK, for each is perpendicular to AB. .". Since CH and DK are equal and parallel, CHKD is a parallelogram. .*. CD is parallel to HK or AB. Q.E.D. THEOREM 19 If a triangle and a parallelogram are on the same base and between the same parallels, the area of the triangle is equal to half that of the parallelogram. C Y X Given the triangle ABC and the parallelogram ABXY on tho same base AB and between the same parallels AB, CX. To Prove AABC = J ||gram ABXY. Join BY. The AS ABC, ABY are on the same base and between the same parallels. .'. AABC = AABY in area. Since the diagonal BY bisects the ||gram ABXY, AABY = \ Hgram ABXY ; .'. AABC - 1 Jlgrani ABXY. Q.E.D. The following formula for the area of a triangle is important : If a, 6, c are the lengths of the sides of a triangle and if s = J(a + 6 + c), the area of the triangle 222 CONCISE OEOMETBY By using the results : Area of parallelogram = height x base, Area of triangle = height x base. Proofs similar to the proof of Theorem 18 can be easily obtained for the following theorems : (1) Triangles on equal bases and between the same parallels are equal in area. (2) Parallelograms on equal bases and between the same parallels are equal in area. (3) Triangles of equal area, which are on equal bases in the same straight line and on the same side of it, are between the same parallels. (4) Parallelograms of equal area, which are on equal bases in the same straight line and on the same side of it, are between the same parallels. (5) The area of a trapezium = the product of half the 'sum of the parallel sides and the distance between them. For riders on Theorems 16, 17, 18, 19, see page 28. THEOREM 20. [PYTHAGORAS' THEOREM.] In any right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the sides containing the right anglp! P FIG. 248. EKOOFS OF THEOREMS 223 Given L BAG is a right angle. To Prove the square on BC = the square on BA 4- the square on AC. Let ABHK, ACMN, BCPQ be the squares on AR, AC, BC. Join CH, AQ. Through A, draw AXY parallel to BQ cutting BC, QP at X, Y. Since L. BAC and L BAK arc right angles, KA and AC are in the same straight line. Again L. HBA - 90 = - QBC. Add to each L. ABC, .'. u HBC= L ABQ. In the As HBC, ABQ HB = AB, sides of square. CB = QB, sides of square. _ HBC^ t- ABQ, proved. /. AHBC-AABQ. Now A HBC and square HA are on the same base HB and between the same parallels H B, K AC ; .'. A HBC=| square HA. Also A ABQ and rectangle BQYX are on the same base BQ and between the same parallels BQ, AXY. .". AABQ - \ rect. BQYX. .'. square HA -- rect. BQYX. Similarly, by joining AP, BM, it can be shown that square M Afreet. CPYX; .'. square HA + square MA = rect. BQYX + rect. CPYX = square BP. Q.E.D. THEOREM 21 If the square on one side of a triangle is equal to the sum of the squares on the other sides, then the angle contained by these sides is a right angle. r. y FIG. 249. 224 CONCISE GEOMETRY Given AB 2 + BC 2 -= AC 2 . To Prove u ABC = 90. Construct a triangle XYZ such that XY-AB, YZ-BC, L. XYZ - 90. Since L XYZ = 90, XZ 2 = XY a + YZ a . But X Y - AB and YZ - BC. /. XZ 2 -AB 2 + BC 2 = AC 2 given. /. XZ-AC. .'. in the A ABC, XYZ AB = XY, constr. BC = YZ, constr. AC = XZ, proved. .'. AABC-AXYZ. .'. L. ABC ~L XYZ. But L XYZ = 90 constr. /. L. ABC 90. For riders on Theorems 20, 21, see page 38. Q.K.D. DEFINITION. If AB and CD are any two vstraight lines, and if AH, BK are the perpendiculars from A, B to CD, then HK is called of AB on CD. FIG. 250(1). Thus, in Fig. 248, QY is the projection of BA on QP, XC is the projection of AC on BC, BX is the projection of QA on BC. Or, in Fig. 250(2), AN is the projection of AC dh AB, BN is the projection of BC on AB. PROOFS OF THEOREMS 225 THEOREM 22 In an obtuse-angled triangle, the square on the side opposite the obtuse angle is equal to the sum of the squares on the sides containing it plus twice the rectangle contained by one of those sides and the projection on it of the other. B AN FIG. 250(2). Given L BAG is obtuse and CN is the perpendicular from C to BA produced. To Prove BC 2 = BA 2 + AC 2 + 2BA . AN. [Put in a small letter for each length that comes in the answer and also for the altitude.] Let BC = a units, B A = c units, AC = b units, AN x units, CN = h units. It is required to prove that a? = c 2 + b 2 + 2cx. Since L BNC = 90, a 2 = (c - Since L ANC = 90, b* = :. a* = or BC 2 = BA 2 + AC 2 + 2BA . AN. Q.E.D. * THEOREM 23 In any triangle, the square on -the side opposite an acute angle is equal to the sum of the squares on the sides containing 15 226 CONCISE GEOMETRY it minus twice the rectangle contained by one of those sides and the projection on it of the other. C X PIG. 5~9(1). FIG. 59(2). Given L BAC is acute and CN is the perpendicular from C to AB or AB produced. To Prove BC 2 = BA 2 + AC 2 - 2AB . AN. [Put in a small letter for each length that comes in the answer and also for the height.] Let BC = a units, BA = c units, AC = b units, AN = x units, CN==A units. It is required to prove that a 2 = c 2 + 6 2 - 2ra. In Fig. 59(1), BNc-#; in Fig. 59(2), BN=#-c. Since L. CNB - 90, a 2 - (c - a?) 2 + A 2 in Fig. 59(1), or a 2 - (x - c) 2 + A 2 in Fig. 59(2) ; / in each case, a 2 c 2 Since u ANC = 90, 6 2 = a? 2 + A 2 , or BC 2 = BA 2 + AC 2 - 2 AB . AN. Q.K.D. THEOREM 24. [APOLLONIUS* THEOREM.] In any triangle, the sum of the squares on two aides is equal to twice the square on half the base phw twice the square on the median which bisects the base. ^ROOFS OF THEOREMS 227 Given D is the mid-point of BC. To Prove AB 2 + AC 2 = 2AD 2 + 2BD 2 . Draw AN perpendicular to BC. From the triangle ADB, AB 2 - AD 2 + DB S From the triangle ADC, AC 2 = AD 2 + DC 2 - 2DC . DM. But BD = DC, given; .'. BD.DN-DC.DN and BD 2 .'. adding, AB 2 + AC 2 = 2AD 2 + 2DB 2 . Q.R.D. For riders on Theorems 22, 23, 24, see page 44. THEOREM 25 (1) If A, B, C, D are four points in order on a straight line, then AC . BD - AB . CD + AD . BC. (2) Tf a straight line AB is bisected at O, and if P is any other point on AB, then AP 2 + PB 2 = 2AO 2 + 2OP 2 . x y ss * FIG. 251(1). (1) Let AB x units, BC = y units, CD ~z units. Then AC = # + y, BD = y + z. .'. AC.BD = (o?-fy) (y-f z) Also AD = x -i- y -f . /. AB . CD + AD . BC = xz + (x + y + z) y .*. AC . BD = AB . CD + AD . BC. "A o > B~ FIG. 251(2). (2) Let AO = # units, OJP^y units. Also PB = OB - OP rr x - y andAP = AO-fOP 228 CONCISE GEOMETRY = 2AO 2 + 2OP 2 . For riders on Theorem 25, see page 46. Q.B.D. GEOMETRICAL ILLUSTRATIONS OF ALGEBRAIC IDENTITIES II. Y M X Fio. 252(1). Draw a line PQ of length a + b inches and take a point R on it such that RQ is of length b inches. On PQ and RQ describe squares PQXY, RQHK on the same side of PQ and produce RK, HK to meet XY, PY at M, L. Then the area of PQXY is (a 4- b)' 2 sq. inches. The areas of LKMY and RQHK are a 2 sq. inches and 6 2 sq. inches. The area of each of the rectangles PK, KX is ab sq. inches. (a + J)(a-6) = tt 2 P R Q FIG. 252(2). PROOFS OF THEOREMS 229 Draw a line PQ of length a inches (a>b) and cut off a part PR of length b inches. On PQ and PR describe squares PQXY, PRHK; produce KH to meet QX at L. Produce KL, YX to E, F so that LE = XF = 6 inches. Now LX = QX-QL = QX-RH=a-6 inches. .*. the rectangle LXFE equals the rectangle HLQR. .". the rectangle KYFE equals the sum of the rectangles KYXL and HLQR equals PQXY - PRHK = a 2 -6* sq. in. But KY = a - b inches, YF = a 4- b inches. THEOREM 26 (1) If two sides of a triangle are unequal, the greater side has the greater angle opposite to it. (2) If two angles of a triangle are unequal, the greater angle has the greater side opposite to it. FIG. 253. (1) GumAOAB To Prove L ABC> L ACB. From AC cut off a part AX equal to AB. Join BX. Since AB = AX, L. ABX = L AXB* But ext. L AXB > int. opp. L. XCB, /. L ABX > L XCB. But 2L.AfeOz.ABX, /. L ABC > L XCB or L ACB. 230 CONCISE GEOMETRY (2) Given L ABC > L ACB. To Prom AOAB. If AC is not greater than AB, it must either be equal to AB, or less than AB. A Fia 61. If AC AB, _ ABC = L- ACB, which is contrary to hypothesis. IfAC<AB, ^.ABC< L ACB, which is contrary to hypothesis. .". AC must be greater than AB. Q.E.D. THEOREM 27 Of all straight lines that can be drawn to a given straight line from an external point, the perpendicular is the shortest. A N P B FIG. 62. Given a fixed point O and a fixed line AB. ON is the perpendicular from O to AB, and OP is any other line from O to AB. To Prove ON<OP. Since the sum of the angles of a triangle is 2 rt. angles, and since ^ONP=1 rt. angle. .'. L NPO + L NOP = 1 rt. angle. L NPO< 1 rt. angle. /. L NPO< - ONP. ON<OP. Q.B.D PROOFS OF THEOREMS 231 THEOREM 28 Any two sides of a triangle are together greater than the third side. x *'' p *V 'I * I * i A-'' C B Fio. 254. Given the triangle ABC. To Prom BA + AOBC, Produce BA to P and cut off AX equal to AC. Join CX. Since AX = AC, L. ACX -= u AXC. But L BCX > L ACX. /. L BCX > L. AXC. .'. in the triangle BXC, ^ BCX > - BXC. .'. BX>BC. But BX = BA + AX = BA + AC. /. BA + AOBC. Q.E.D. The following theorem is an easy rider on the above : The shortest and longest distances from a point to a circle lie along the diameter through the point. FIG. 255. If AB is a diameter, tod If P lies on AB produced, PA>PQ>PB. 232 CONCISE GEOMETRY Join Q to the centre O. PA = PO + OA-PO l-OQ>PQ. PB + BO=PO<PQ + QO. For riders on Theorems 26, 27, 28 see page 49. THEOREM 29 The straight line joining the middle points of two sides of a triangle is parallel to the base and equal to half the base. A C FIG. 256. Given H, K are the middle points of AB, AC. To Prove HK is parallel to BC and HK = BC. Through C, draw CP parallel to BA to meet HK produced at P. In the As AHK, CPK. /.AHK- -CPK, alt. _ s. i_HAK^ 1.PCK, alt. L s. AK = KC, given. .'. AAHK-ACPK. /. CP = AH. But AH = BH, given. .'. CP = BH. Also CP is drawn parallel to BH. /. the lines CP, BH are equal and parallel. .". BCPH is a parallelogram. .'. HK Is parallel to BC. Also HK = KP from congruent triangles. But HP = BC oppf sideft of parallelogram. .'. HK-JBC. Q.E.D. PROOFS OF THEOREMS 233 THEOREM 30 If there are three or more parallel straight lines, and if the inter- cepts made by them on any straight line cutting them are equal, then the intercepts made by them on any other straight line that cuts them are equal. /\ H \ D/ f \ 8 E/ K \T KKJ. 257. Given three parallel lines cutting a line AE at B, C, D and any other line PT at Q, R, S and that BC = CD. 1 To Prove QR^RS. Draw BH, CK parallel to PT to meet OR, DS at H, K. Then BH is parallel to CK. .'. in the AS BCH, CDK. L CBH ^ L DCK corresp. - a. L BCH = L CDK corresp. _ s. BC=^CD, given. .'. ABCH = ACDK. /. BH-CK, But BQRH is a ||gram since its opposite sides are parallel. .'. BH=QR. And CRSK is a ||gram since its opposite sides are parallel. :. CK-RS. /. QR = RS. t Q.E.D. For riders on Theorems 29, 30 see page 52. 234 CONCISE GEOMETRY BOOK III THKORKM 31 (1) The straight line which joins the centre of a circle to the middle point of a chord (which is not a diameter) is per- pendicular to the chord. (2) The line drawn from the centre of a circle perpendicular to a chord bisects the chord. FIG. 69. (1) Given a circle, centre O, and a chord AB, whose mid-point is N. To Prove u ONA is a right angle. Join OA, OB. In the As ONA, ONB, OA = OB, radii. AN = BN, given. ON is common, /. A ONA = A ONB. /. i.ONA= -LONB. But these are adjacent angles, .". each is a right angle. (2) Given that ON is the perpendicular from the centre O of a circle to a chord AB. To Prove that N is the mid-point of AB. In the right-angled triangles ONA, ONB. OA = OB, radii. ON is common. /. A ONA = A ONB. /. AN = NB. * Q.E.D. PJ800FS OF THEOREMS 235 THKOBEM 32 In equal circles or in the same circle : (1) Equal chords are equidistant from the centres. (2) Chords which are equidistant from the centres are equal. FIG. 258(1). (1) Given two equal circles ABX, CDY, centres P, Q, and two equal chords AB, CD. To Prove that the perpendiculars PH, QK from P, Q to AB, CD are equal. Join PA, QC. Since PH, QK are the perpendiculars from the centres to the chords AB, CD, H and K are the mid- points of AB and CD. /. AH = JAB and CK = 'CD. But AB = CD, given. /. AH = CK. .*. in the right-angled triangles PAH, QCK, the hypotenuse PA = the hypotenuse QC, radii of equal circles. AH = CK, proved. .'. A PAH 53 A QCK. .'. PH = QK. Q.E.D. (2) Given that the perpendiculars PH, QK from P, Q to the chords AB, CD are equal To Prove that AB = CD. In the right-angled triangles PAH* QCK, the hypotenuse PA = the hypotenuse QC, radii of equal circles. PH =* QK, given. * .'. A PAH B A QCK. * /. AH-CK. 236 CONCISE GEOMETRY But the perpendiculars PH, QK bisect AB, CD. .'. AB = 2AH and CD = 2CK. .'. AB = CD. Q.E.D. The proof is unaltered if the chords are in the same circle. FIG. 258(2). For riders on Theorems 31, 32, see page 57. THEOREM 33 The angle which an arc of a circle subtends at the centre is double that which it subtends at any point on the remaining part of the circumference. FIG. 259(1). FIG. 259(2). /N FIG. 259(3). Given AB is an arc of a circle, centre O ; P is any point on the remaining part of the circumference. To Prove L AOB = 2 L APB. Join PO, and produce it to any point N. Since OA = P, L CAP = L OPA. But ext. Z-NOA = int. L CAP 4- int. Z.OPA. PROOFS OF THEOREMS 237 Similarly L NOB - 2 L OPB. /. adding in Fig. 259(1) and subtracting in Fig. 259(2), we have L AOB = 2 L APB. Q.E.D. Fig. 259(3) shows the case where the angle AOB is reflex, i.e. greater than 180 : the proof for Fig. 259(3) is the same as for Fig. 259(1). THEOREM 34 1) Angles in the same segment of a circle are equal. 2) The angle in a semicircle is a right angle. B- Q 'B FIG. 76(1). Fio. 76(2). 1) Given two angles APB, AQB in the same segment of a circle. To Prove L APB = L AQB. Let O be the centre. Join OA, OB. Then ^ AOB - 2 L APB. ^ at centre = twice L at Qce. and <_ AOB = 2 L AQB. .'. _ APB = L AQB. Q.E.D. 2) Given AB a diameter of a circle, centre O, and P a point on the circumference. To Prove L APB = 90. Fm. 77. 238 CONCISE GEOMETRY L. AOB = 2 L APB. L at centre twice L at O ce Bat _ AOB *= 180, since AOB is a straight line ; /. L APB = 90. Q.K.D. THEOREM 35 (1) The opposite angles of a cyclic quadrilateral are supplementary. (2) If a side of a cyclic quadrilateral is produced, the exterior angle is equal to the interior opposite angle. (1) Oiven ABCD is a cyclic quadrilateral. To Prove L ABC + L ADC = 180. Let O be the centre of the circle. Join OA, OC. Let the arc ADC subtend angle X Q at the centre, and let the arc ABC subtend angle y at the centre. Now # 2 L ABC. L. at centre = twice L at and y = 2 ^ ADC. /. 2 L ABC + 2 L ADC = 360. .'. L ABC + /L ADC = 180. Q.B.D. FIG. 78. PROOFS OF THEOREMS 239 (2) Given the side AD of the cyclic quadrilateral ABCD i* pro- duced to P. To Prove L PDC =* L ABC, Now L ADC 4- L PDC = 180, adj. angles, and L ADC -f L ABC - 180, opp. L s cyclic quad. .'. L ADC + L PDC = L. ADC + L ABC. .'. L PDC ==> L ABC. Q.E.D. For riders on Theorems 33, 34, 35 see page 62. THEOREM 36 (1) If the line joining two points subtends equal angles at two other points on the same side of it, then the four points lie on a circle. (2) If the opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic. FIG. 261. (1) Given that L APB = L AQB where P, Q are points on the same side of AB. To Prove that A, P, Q, B lie on a circle. If possible, let the circle through A, B, P not pass through Q and let it cut AQ or AQ produced at X. Join BX. Then L AXB = L. APB, same segment, and t- AQB = L APB, given. u AXB = L AQB. that is, the exterior angle of the triangle BQX equals the interior opposite angle, whicbjs impossible* /. the circle through A,-B, P must pass through Q, Q.RD. 240 CONCISE GEOMETRY (2) Given that in the quadrilateral ABCD, L ABC + L ADC = 180. To Prove that A, B, C, D lie on a circle. If possible let the circle through A, B, C not pass through D, and let it cut AD or AD produced at X. Join CX. .D Flu. 2G2. Then L ABC + L AXC = 180, opp. L s cyclic quad. But L ABC + L ADC - 180, given. /. L AXC - L ADC. That this, the exterior angle of the triangle CXD equals the interior opposite angle, which is impossible, .". the circle through A, B, C must pass through D. Q.E.D. For riders on Theorem 36, see page 83. THEOREM 37 In equal circles (or in the same circle), if two arcs subtend equal angles at the centres or at the circumferences, they are equal. Given two equal circles, ABP, CDQ, centres H, K. (1) Given that L AHB = L CKD. To Prove that arc AB = arc CD PKOOFS OF THEOREMS 241 Apply the circle AB to the circle CD so that the centre H falls on the centre K and HA along KG. Since the circles are equal, A falls on C and the circum- ferences coincide. Since - AHB = _ CKD, HB falls on KD, and B falls on D. the arcs AB, CD coincide, arc AB = arc CD. (2) Given that _ APB = L CQD. To Prove that arc AB = arc CD. Now L. AHB ^= 2 L. APB, - at centre = twice - at Oce. and L CKD = 2 L CQD. But L APB = L. CQD, given. *. AHB = L. CKD. arc AB = arc CD. Q.E.I). THEOREM 38 In equal circles (or in the same circle), if two arcs are equal, they subtend equal angles at the centres and at the circumferences. P Given two equal circles ABP, CDQ, centres H, K, and two equal arcs AB, CD. To Prove (1) L. AHB = L CKD. (2) z.APB-iCQD. (1) Apply the circle AB to the circle CD so that the centre H falls on the centre K and HA along K9 Since the circles are equal, A falls on C and the circum- ferences coincide. But arc AB = arc CD, /. B falls' on D and HB on KD. /. _ AHB coincides with" L CKD. .'. - AHB L. CKD. Q.E.D. 16 242 CONCISE GEOMETRY (2) =<_ CKD. But L. AHB = L CKD, just proved. .'. L APB = L. CQD. Q.E.D. THEOREM 30 In equal circles or in the same circle (1) if two chords are equal, the arcs which they cut off are equal (2) if two arcs are equal, the chords of those arcs are equal. Given two equal circles ABP, CDQ, centres H, K. (1) Given chord AB = chord CD. To Prove arc AB = arc CD. Join HA, HB, KC, KD. In the As HAB, KCD, HA ~ KC, radii of equal circles. HB = KD, radii of equal circles. AB = CD, given. * .'. AHAB-AKCD. /. ^.AHB= Z-CKD. .". the arcs AB, CD of equal circles subtend equal angles at the centres. .*. arc AB = arc CD. Q.E.D. (2) Given arc AB = arc CD. To Prove chord AB =fchord CD. r Since AB, CD are equal arcs of equal circles, = LCKD. PKOOFS OF THEOREMS /. in the AS HAB, KCD, H A = KC, radii of equal circles. HB = KD, radii of equal circles. z.AHB= Z.CKD, proved. AB-CD. For riders on Theorems 37, 38, 39 see page 72. 243 Q.E.D. THE TANGENT TO A CIRCLE FIG. 265. Let P be any point on an arc AB of a circle. Suppose a point Q starts at A and moves along the arc AP towards P, taking successive positions Q lf Q 2 , Q 8 . . . and draw the lines PQ V PQ 2 , PQ 3 . . . Also suppose a point R starts at B and moves along the arc BP towards P, taking successive positions R 1? R 2 , R 3 . . . and draw the lines PR 1} PRg, PR 8 . . . All lines in the PQ system cut off arcsalong PA, the lengths of which decrease without limit as Q tends to P. All lines in the PR system cut off gjrcs along PB, the lengths of which decrease without limit as R tends to P. Produce AP, BP to X, Y. All lines drawn from P in the angle APY or BPX belong either 244 CONCISE GEOMETRY to the PQ system or to the PR system, except t/ie single line which cuts off an arc of zero length. This line is called the tawjent at P. The tangent at P is therefore the lino CPD, which is the inter- mediate position between lines of the PQ system and lines of the PR system, arid cuts off an arc of zero length at P. THEOREM 40 The tangent to a circle is at right angles to the radius through the point of contact. FIG. 267(1). FIG. 267(2). Given P is any point on a circle, centre O. To Prove the tangent at P is perpendicular to OP. Through P, see Pig. 267(1), draw any line XPQY, cutting the circle again at Q. Join OP, OQ. t OP OQ, radii, .'. L OPQf= L OQP. .*. their supplements are equal, ,\ <_ OPX L OQY. PROOFS OF THEOREMS 245 Now the tangent at P is the limiting position of the line XPQY, when the arc PQ is decreased without limit, so that Q coincides with P, see Fig. 267(2). .'. in Fig. 267(2), z_OPX = 2LOPY. But these are adjacent angles, .*. each is a right angle. /. in Fig. 267(2), L. OPX 90, where PX is the tangent at P. Q.E.D. THEOREM 41 If a straight line touches a circle and, from the point of contact, a chord is drawn, the angles which the chord makes with the tangent are equal to the angles in the alternate segments of the circle. K Y/ "V Fui. 268(1). FIG. 268(2). Given YPX is a tangent at P to the circle PLAK, and PA is any chord through P. To Prove L APX = L PKA and L APY L PLA. In Fig. 268(1), draw through P any line YPQX cutting the circle again at Q. Join QA. Then L AQX = L PKA ; ext. L of cyclic quad. = int. opp. /. . Now the tangent at P is the limiting position of the line YPQX when the arc PQ is decreased without limit, so that Q coincides with P, see Fig. 268(2). But the limiting position of /. AQX is /. APX. .". when YPQX becomes the tangent at P, Similarly it may be} proved that Z. APY= L PLA. Q.E.D. 246 CONCISE GEOMETRY The converse of this theorem is frequently of use in rider- work. For riders on Theorems 40, 41, see page 68. THEOKEM 42 If two tangents are drawn to a circle from an external point (1) The tangents are equal. (2) The tangents subtend equal angles at the centre. (3) The line joining the centre to the external point bisects the angle between the tangents. T FIG. 86. Given TP, TQ are the tangents from T to a circle, centre O. To Prove (I) TP-TQ. (2) (3) Since TP, TQ are tangents at P, Q, the angles TPO, TQO are right angles. .". in the right-angled triangles TOP, TOQ OP = OQ, radii. OT is the common hypotenuse. and ZTOP=ZTOQ, and Z.OTP = ZOTQ. Q.E.D. THEOREM 43 t If two circles touch one another* the line joining their centres (produced if necessary) passes through the point of contact. PROOFS OF THEOREMS 247 Given two circles, centres A, B, touching each other at P. To Prove AB (produced if necessary) passes through P. Fio. 87(2). Since the circles touch each other at P, they have a common tangent XPY at P. Since XP touches each circle at P, the angles XA, XPB are right angles. .". A and B each lie on the line through P perpendicular to PX. .". A, B, P lie on a straight line. Q.E.D. Note. If two circles touch each other externally (Fig. 87(1)), the distance between their centres equals the sum of the radii. If two circles touch each other internally (Fig. 87(2)), the distance between their centres equals the difference of the radii. For riders on Theorems 42, 43, see page 77. THEOREM 44 In a right-angled triangle, the line joining the mid-point of the hypotenuse to the opposite vertex is equal to half the hypotenuse. Given ABC is a triangle, right-angled at A, and D is the mid- point of BC. 248 CONCISE GEOMETRY To Prow AD Draw a circle through A, B, C. Since /. BAG = 90, BC is a diameter. FIG. 269. But D is the mid-point of BC, .". D is the centre of the circle. DA = DB = DC, radii. .'. DA-JBC. Q.E.T). DEFINITION. If a point moves in such a way that it obeys a given geometrical condition, the path traced oat by the point is called the locus of the point. THEOREM 45 The locus of a point, which is equidistant from two given points, is the perpendicular bisector of the straight line joining the given points. A N B , FIG. 105. Given two fixed points A, B and any position of a point P which moves so that PA = PB. To Prove that P lies on the 'perpendicular bisector of AB. Bisect AB at N. Join PN. PROOFS OF THEOREMS 249 In the AS ANP, BNP, AN = BN, constr. AP = BP, given. PN is common. .'. AANPsABNP. .'. L ANP = Z. BNP. But these are adjacent angles, .*. each is a right angle. PN is perpendicular to AB and bisects it. .*. P lies on the perpendicular bisector of AB. Q.E.D. THEOREM 46 The locus of a point which is equidistant from two given inter- secting straight lines is the pair of lines which bisect the angles between the given lines. Fin. 106. Given two fixed lines AOB, COD and any position of a point P which moves so that the perpendiculars PH, PK from P to AOB, COD are equal. To Prove P lies on one of the two lines bisecting the angles BOC, BOD. Suppose P is situated in the angle BOD. In the right-angled triangles PHO, PKO, PH = PK, given. PO is the common hypotenuse. /. A PHO s A PKO. /. L POH = L ROK. .". P lies on the line bisecting the angle BOD. In the same way if P is situated in either of the angles BOC, COA, AOD, it Jj#s on the bisectors of these angles. For riders on Theorems 45, 46, see page 94. Q ' B ' D ' 250 CONCISE GEOMETRY THEOREM 47 The perpendicular bisectors of the three sides of a triangle are concurrent (i.e. meet in a point). Given that the perpendicular bisectors OY, OZ of AC, AB meet at O. To Prove the perpendicular bisector of BC passes through O. Bisect BC at X, join OX ; also join OA, OB, OC. In the As OZA, OZB, BZ ZA, given. OZ is common. /. BZO = /. AZO, given rt. </ s. /. A OZA -A OZB. .'. OA-OB. Similarly from the AH OYA, OYC, it can be proved that OA = OC, .'. OB - OC. In the As OXB, OXC, OB = OC, proved. XB = XC, constr. OX is common. /. AOXB == AOXC. /. L OXB -/. OXC. But these are adjacent angles, .". each is a rt. _ . .*. OX is the perpendicular bisector of BC. ^ Q.E.D. For riders on Theorem 47, see page 99. PROOFS OF THEOREMS 251 THEOREM 48 The internal bisectors of the three angles of a triangle are concurrent. A B PC FIG. 109. Given that the internal bisectors IB, 1C of the angles ABC, ACB meet at I. To Prove that I A bisects the angle BAC. Join I A. Draw IP, IQ, IR perpendicular to BC, CA, AB. In the As IBP, IBR, ZJBP= Z.IBR, given. ZIPB= Z.IRB, constr. rt. Z.s. IB is common. /. IP = IR. Similarly from the AS ICP, ICQ it may be proved that IP = IQ, .'. IQ = IR. In the right-angled triangles IAQ, IAR, IQ = IR, proved. IA is the common hypotenuse. .'. ZJAQ=Z.IAR. IA bisects the angle BAC. Q.E.D. For riders on Theorem 48, see page M)0. THEOREM 49 The three altitudes of a triangle (i.e. the lines drawn from the vertices perpendicular to the opposite sides) are concurrent. 252 CONCISE OEOMETBY Given AD, BE, OF are the altitudes|of the triangle ABC. To Prove AD, BE, CF are concurrent. Through A, B, C draw lines parallel to BC, CA, AB to form the triangle PQR. v P FIG. 270. Since BC is || to AR and AC is || to BR, BCAR is a parallelogram. /. BC^AR. Similarly, since BCQA is a parallelogram, BC = AQ, .'. AR = AQ. Since AD is perpendicular to BC, and since QR, BC are parallel, .'. AD is perpendicular to QR. But AR = AQ, .*. AD is the perpendicular bisector of QR. Similarly, BE and CF are the perpendicular bisectors of PR, PQ. But the perpendicular bisectors of the sides of the triangle PQR are concurrent .*. AD, BE, CF are concurrent. Q.E.D. For riders on Theorem 49, see page 101. THEOREM 50 (1) The three medians of a triangle (i.e. the lines joining each vertex to the middle point of the opposite side) are con- current. PROOFS OF THEOREMS 253 (2) The point at which the medians intersect is one-third of the way up each median (measured towards the vertex). (1) Given the medians BE, CF of the triangle ABC, intersect at G. To Prove that AG, when produced, bisects BC. Join AG and produce it to H, so that AG - GH. Let AH cut BC at D ; join HB, HC. Since AF = FB and AG = GH, FG is parallel to BH. Since AE = EC and AG = GH, EG is parallel to CH. Since FGC and EGB are parallel to BH and CH, BGCH is a parallelogram ; .*. the diagonals BC, GH bisect each other ; /. BD-DC. Q.E.D. (2) For the same reason, GD = DH. But AG*=GH. C. AG-2GD. /. AD-3GD. or GD For riders on Theorem 50* see page 103. 254 CONCISE GEOMETRY BOOK IV THEOREM 51 If two triangles have equal heights, the ratio of their areas is equal to the ratio of their bases. A X Y FIG. 113. Given two triangles ABC, XYZ having equal heights AP, XQ. A ABC BC = The area of a triangle = height x base. .'. AABC=JAP.BC. and AXYZ-=JXQ.YZ, AABC = |AP.BC " A"XYZ""|XQ7YZ' But AP = XQ, given, . AABC = BC " A XYZ Yr Q.E.D. THEOREM 52 (1) If a straight line is drawn parallel to one side of a triangle, it divides the other sides (produced if necessary) proportionally. (2) If a straight line divides two sides of a triangle proportionally, it is parallel to the third side. H K FIG. 114(2). B O FIG. 114(3). PROOFS OF THEOREMS 255 (1) Given a line parallel to BC cuts AB, AC (produced if necessary) at H, K. ni AH AK To Prove = ----- HB KC Join BK, CH. The triangles KHA, KHB have a common altitude from K to AB. . AKHA^AH " AKHB~HB' The triangles HKA, HKC have a common altitude from H to AC. . AHKA = AK A H KC ~ KG* But A KHB, AKHC are equal in area, being on the same base HK and between the same parallels HK, BC. AH_AK " HB'KC' (2) Given a line HK cutting AB, AC at H, K such that AH = AK . V ' b HB KC To Prove HK is parallel to BC. The triangles KHA, KHB have a common altitude from K to AB. . AKHA^AH AKHB HB' The triangles HKA, HKC have a common altitude from H to AC. . AHKA^AK AHKC~~KC' u . AH AK . J3ut - = . given. HB KG' b . AKHA _ AHKA But these triangles are on the same base HK and on the same side of it. * .*. HK is parallel to BC. Q.B.D. 256 CONCISE GEOMETRY COROLLARY 1. If a line HK cuts AB, AC at H, K so that AHAK Now AH AK d HB KC AB^AC a ' IU AB~AC' *AK. . HB + AI H KC + AK "*" HB =1 AB AC KC' HB KC HB~KC' HB KC AB^AC' Q. A1 HB AH KC AK A1S AB X HB == AC X iTc AH_AK AB~AC* COROLLARY 2. If a lino HK parallel to BC cuts AB, AC at H, K, rn, AH AK , HB KC Then = - and - = AB AC AB AC COROLLARY 3. If a line HK cuts AB, AC at H, K so that A - K , then HK is parallel to BC. AB AC For riders on Theorems 51, 52 see page 106. THEOREM 53 If two triangles are equiangular, their corresponding sides are proportional. A X B C FIG. 272. .PROOFS OF THEOREMS 257 Given the triangles ABC, XYZ are equiangular, having Z.A XY XZ YZ From AB, AC cut off AH, AK equal to XY, XZ. Join HK. In the As AHK, XYZ, AH = XY, constr. AK - XZ, constr. ZHAK- Z.YXZ, given. /. AAHK-AXYZ. But /.XYZ = ,/ ABC, given. .'. Z.AHK=Z.ABC. But these are corresponding angles, .". HK is parallel to BC. . AB_AC " AH~AK* But AH - XY and AK = XZ. . A]3 = AC XY ~ XZ' AC BC Similarly it can be proved that = - . DEFINITION. If two polygons are equiangular, and if their corresponding sides are proportional, they are said to be similar. Theorem 53 proves that equiangular triangles are necessarily similar. THEOREM 54 If the three sides of one triangle are proportional to the three sides of the other, then the triangles are equiangular. X Y, c Fio. 273. P 258 CONCISE GEOMETRY Given the AS ABC, XYZ are such that - --- = ~y' /v T i z. 2./V ^o Prove Z.A= Z.X, ZB= Z.Y, Z.C = Z.Z. On the side of YZ opposite to X, draw YP and ZP so that /.ZYP- Z ABC and Z.YZP= ZACB. Since the AS ABC, PYZ are equiangular, by construction, AB_BC YP~YZ' AB _ AB VD W . T r AT. Similarly ZP = XZ. /. in the AS XYZ, PYZ. XY-PY, proved. XZ = PZ, proved. YZ is common. AXYZ^APYZ. ZLXYZ- ZPYZ and Z.XZY=ZPZY. But Z.PYZ- Z.ABC and /.PZY= Z.ACB, constr. L XYZ = z. ABC and L XZY = L ACB. /. also Z. YXZ = Z BAG. Q.E D. THEOREM 55 If two triangles have an angle of one equal to an angle of the other, and the sides about these equal angles proportional, the triangles are equiangular. HA. XK PROOFS OF THEOREMS 259 Given in the triangles ABC, XYZ, Z.BAC- Z.YXZ and . j\ i AC XZ' To Prove /.ABC = Z.XYZ and Z.ACB = Z.XZY. From AB, AC, cut off AH, AK equal to XY, X7. Join HK. In the As AHK, XYZ, AH = XY, constr. AK = XZ, constr. Z.HAK- ZYXZ, given. AAHK-AXYZ. /.AHK- Z. XYZ and /AKHr-^XZY. Now A B ^ AC d XY ^ AH, XZ - AK. XY XZ AB_AC AH ~ AK' .". HK is parallel to BC. L AHK = /. ABC and Z AKH = /_ ACB, corrosp. /. s. Miit ZAHK - Z.XYZ and _ AKH - Z.XZY, proved. L ABC - L XYZ and L ACB = L XZY. For riders on Theorems 53, 54, 55 see page 1J2. Q- K - !) - THEOREM 56 (1) If two chords of a circle (produced if necessary) cut one another, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other. (2) If from any point outside a circle, a secant and a tangent are drawn, the rectangle contained by the whole secant and the part of it outside the circle is equal to the square on the tangent. 129(1). Fio. 129(2). 260 CONCISE GEOMETRY (1) Given two chords AB, CD intersecting at O. To Prove OA . OB = OC . OD. Join BC, AD. In the As ADD, BOG, Z.OAD= Z.OCB, in the same segment, Fig. 129(1) and Fig 129(2). ZAOD= ,/ COB, vert. opp. in Fig. 129(1), same L in Fig. 129(2). .'. the third L ODA = the third L OBC. .". the triangles are equiangular. OA_OD 6C~OB* /. OA.OB = OC.OD. Q.E.D. (2) Given a chord AB meeting the tangent at T in O. Join AT, BT. FIG. 180. In the As AOT, JOB, ^TAO= Z.BTO, alt. segment. Z.AOT= Z.TOB, same angle. .'. the third ,/ ATO = the third Z.TBO. /. the triangles are equiangular. OA^OT, OT ~ OB' /. OA.OB = OT 2 . Q.E.D. Note. This may also be deduced from (1) by taking tlje limiting case when D coincides with C in Fig. 129(2). PKOOFS OF THEOREMS 261 The converse properties are as follows : (i) If two lines AOB, COD are such that AO . OB -= CO . OD, then A, B, C, D lie on a circle. (ii) If two lines OBA, ODC are such that OA . OS = OC . OD, then A, B, C, D lie on a circle. (iii) If two lines OBA, OT are such that OA . OB = OT 2 , then the circle through A, B, T touches OT at T. These are proved easily by a reductio ad absurdum method. THEOREM 57 If AD is an altitude of the triangle ABC, which is right-angled at A, then (i) AD 2 =BD . DC. (ii) BA 2 =BD . BC. A B C- (1) Since /. BDA = 90, the remaining angles of the triangle ABD add up to 90. /. L DAB 4- Z DBA = 90. But L DAB + L DAC = 90, given. /. L DAB + L DBA = L DAB + L DAC. .'. Z.DBA=/.DAC. /. in the AS ADB, CDA, Z.ADB= /.CDA, right angles. L DBA = Z. DAC, proved. .'. the third L BAD = the third /. ACD. .". the triangles are equiangular. AD_BD DC"" DA' * BD'.DC. Q.E.D. 262 CONCISE GEOMETRY (2) In the A ADB, CAB, L ADB= Z.CAB, right angles. Z.ABD= Z.CBA, same angle. .'. the third /. DAB - the third L ACB. the triangles are equiangular. AB_BD BC~AB .'. AB 2 =BD . BC. Q.E.D. An alternative method of proof is given on page 121. Note. AD is called the mean proportional between BD and DC. Also BA is the mean proportional between BD and BC. For riders on Theorems 56, 57 see page 122. THEOREM 58 The ratio of the areas of two similar triangles is equal to the ratio of the squares on corresponding sides. B H C Y K Z FIG. 133. Given the triangles ABC, XYZ are similar. . A XYZ YZ 2 Draw the altitudes AH, XK. In the As AHB, XKY, Z.ABH=ZXYK, given. ZAHB= Z.XKY, rt. Z.s constr. .'. the third L BAH - the third L YXK. /. the As ArtB, XKY are similar. . AH_AB " XK~KY AB BC * But = , since AS ABC, XYZ are similar. PROOFS OF THEOREMS . AH_BC " XK~YZ" But AABC-^AH . BC and AXYZ-^XK . YZ. A ABC AH . BC ' ' AXYZ ~ XK . YZ' But AH BC AABC_BC 2 Q.E.D. If two polygons are similar, it can be proved that they can be divided up into the same number of similar triangles. Hence it follows that the ratio of the areas of two similar polygons is equal to the ratio of the squares on corresponding sides. THEOREM 59 If three straight lines are proportionals, the ratio of the area of any polygon described on the first to the area of a similar polygon described on the second is equal to the ratio of the first line to the third line. A B C D E F FIG. 134. Given three lines AB, CD, EF such that = and two similar figures ABP, CDQ. m _ figure ABP AB To Prove ^ ~~ = == figure CDQ EF Since the figures are similar, , figure ABP ^AB 8 figure CDQ ""CD 2 ' 264 CONCISE GEOMETRY But CD 2 = AB . EF, given. AB 2 *_ AB 2 AB CD 2 ~AB7EF~EF' figure ABP_ AB figure CDQ~EF' Q.E.D. For riders on Theorems 58, 59 see page 1127. THEOREM 60 (1) If the vertical angle of a triangle is bisected internally or externally by a straight line which cuts the base, or the base produced, it divides the base internally or externally in the ratio of the other sides of the triangle. (2) If a straight line through the vertex of a triangle divides the base internally or externally in the ratio of the other sides, it bisects the vertical angle internally or externally. D C B C FIG. 275(1). FIG. 275(2). (1) Given the line AD bisecting the angle BAC, internally in Fig. 275(1), externally in Fig. 275(2), meets BC or BC produced at D. - ?> BD BA To Prove ^~ c . Through C draw CP parallel to DA to meet AB or AB pro- duced at P. BA is produced to E in Fig. 275(2). In Fig. 275(1). ZBAD= ZAPC, corresp. /.s. Z.DAC= L AGP, alt. /.s. But BAD = /_ DAC, given. Z.APC-Z.ACP. In Fig. 275(2). Z.EAD = Z.APC, corresp. Z.s. Z.DAC*= Z.ACP, alt. L*. PROOFS OF THEOREMS 265 But Z. EAD = Z DAC, given. Z.APC-ZACP. in each case, AP = AC. But CP is parallel to DA. BA _ BD AP "" DC' Q.B.D. BA BD 2) Given that AD cuts BC or BC produced so that ' L AC DC To Prove that AD bisects Z. BAC internally or externally. Through C draw CP parallel to DA to meet AB or AB pro- duced at P. XT ^ n i BA BD Now by parallels. = - . J L AP DC T> i. BA BD But = , BA BA * ' AP ~~ AC* /. AP = AC. .*. Z.APC-ZACP. In Fig. 275(1) Z.APC-- Z.BAD, corresp. Z_s. But Z. APC = Z. ACP, proved. .'. Z BAD - Z DAC. In Fig. 275(2) Z. APC = ZEAD, corresp. Z.s. ZACP= Z.DAC, alt. Z.. 1 But Z. APC - Z ACP, proved. .'. Z. EAD = /_ DAC. /. in Fig. 275(1), AD bisects Z.BAC internally, and in Fig. 275(2), AD bisects Z.BAC externally Q.E.D. For riders on Theorem 60 see page 152. CONSTRUCTIONS FOR BOOK I CONSTRUCTION 1 From a given point in a given straight line, draw a straight line making with the given line an angle equal to a given angle. Q 1 Z A FIG. 276. Given a point A on a given line AB and an angle XYZ. To Construct a line AC such that Z.CAB= Z XYZ. With centre Y and any radius, draw an arc of a circle cutting YX, YZ at P, Q. With centre A and the same radius, draw an arc of a circle EF, cutting AB at E. With centre E and radius equal to QP, describe an arc of a circle, cutting the arc EF at F. Join AF and produce it to C. Then AC is the required line. Proof. Join PQ, EF. In the As PYQ, FAE, YP = AF, constr. YQ = AE, constr. PQ = EF, constr. </XYZ=/.CAEf. Q.E.F. 267 268 CONCISE GEOMETRY CONSTRUCTION 2 Bisect a given angle. Fi. 277. Given an angle BAG. To Construct a line bisecting the angle. With A as centre and any radius, draw an arc of a circle, cutting AB, AC at P, Q. With centres P, Q and with any sufficient radius, the same for each, draw arcs of circles, cutting at R. Join AR. Then AR is the required bisector. Proof. Join PR, QR. In the As APR, AQR, AP = AQ, radii of the same circle. PR = QR, radii of equal circles. AR is common. Q.E.F. ,/PAR=/.QAR. CONSTRUCTION 3 Draw the perpendicular bisector of a given finite straight line. FIG. 278. CONSTRUCTIONS FOR BOOK I 269 Given a finite line AB. To Construct the line bisecting AB at right angles. With centres A, B and any sufficient radius, the same for each, draw arcs of circles to cut at P, Q. Join PQ and let it cut AB at C. Then C is the mid-point of AB, and PCQ bisects AB at right angles. Proof. Join PA, PB, QA, QB. In the As PAQ, PBQ, PA = PB, radii of equal circles. QA = QB, radii of equal circles. PQ is common. .'. APAQ-APBQ. /. Z.APQ=Z.BPQ. In the As APC, BPC, PA = PB, radii of equal circles. PC is common. L APC = Z. BPC, proved. .'. AAPC = ABPC. AC^CB. and /. ACP= Z.BCP. But these are adjacent angles, .*. each is a right angle. Q.E.P. CONSTRUCTION 4 Draw a straight line at right angles to a given straight line from a given point in it. A 'P C /Q B Fio. 279. Given a point C on a line AB. To Construct a line from C perpendicular to AB. With centre C and any* radius, draw an aroof a circle cutting AB at P, Q. 270 CONCISE GEOMETRY With centres P, Q and any sufficient radius, the same for each, draw arcs of circles to cut at R. Join CR. Then CR is the required perpendicular. Proof. Join PR, QR. In the As RCP, RCQ, RP = RQ, radii of equal circles. CP = CQ, radii of the same circle. CR is common. .'. A RCP = A RCQ. /. L RCP -Z. RCQ. But these are adjacent angles, .*. each is a right angle. Q.K.F. CONSTRUCTION 5 Draw a perpendicular to a given straight line of unlimited length from a given point outside it. FIG. 280. Given a line AB and a point C outside it. To Construct a line from C perpendicular to AB. With C as centre and any sufficient radius, draw an arc of a circle, cutting AB at P, Q. With P, Q as centres and any sufficient radius, the same for each, draw arcs of circles, cutting at R. Join CR and let it cut AB at X. Then-*X is perpendicular t<? AB. Proof. Join CP, CQ, RP, RQ. CONSTRICTIONS FOK BOOK I 271 In the As CPR, CQR, CP = CQ, radii of the same circle. RP ~ RQ, radii of equal circles. CR is common. In the AS CPX, CQX, CP = CQ, radii. CX is common. ZPCX- ^QCX, proved. .'. ACPX == ACQX. But these are adjacent angles, .". each is a right angle. Q.E.P. CONSTRUCTION 6 'hrough a given i>oint, draw a straight line parallel to a given straight line. J 9 Fro. 281. liven a line AB and a point C outside it. To Construct a line through C parallel to AB. With C as centre and any sufficient radius, draw an arc of a circle PQ, cutting AB at P. With P as centre and the same radius, draw an arc of a circle, cutting AB at R. With centre P and radius equal to CR, draw an arc of a circle, cutting the arc PQ at Q on the same side of AB as C. Join CQ. Then CQ is parallel to AB. Proof. Join CR, CP, PQ. In the As CRP, PQC, CR = PQ, constr. RP =* QC radii'of equal circles. PC is common. 272 CONCISE GEOMETRY .'. Z.CPR-Z.PCQ. But these are alternate angles, .". CQ is parallel to RP. Q.E.F. CONSTRUCTION 7 Draw a triangle having its sides equal to three given straight lines, any two of which are together greater than the third side. -\(1 id A Bj X FIG. 282. Given three lines a, b y c. To Construct a triangle whose sides are respectively equal to a, by c. Take any line AX, and with A as centre and radius equal to c, draw an arc of a circle, cutting AX at B. With A as centre and radius equal to 6, draw an arc of a circle; and with B as centre and radius equal to a, draw an arc of a circle, cutting the former arc at C. Join AC, BC. Then ABC is the required triangle. Proof. By construction, AB = c. Q.E.F. CONSTRUCTION 8 Draw a triangle, given two angles and the perimeter. Given two angles X, Y and a* line HK. To C&istruct a triangle having two of its angles equal to X and Y and its perimeter equal to HK. CONSTRUCTIONS FOR BOOK I 273 Construct lines PH, QK on the same side of HK such that Z.PHK= Z.X and ZQKH= Z.Y. Construct lines HA, KA intersecting at A and bisecting the angles PHK, QKH. Construct through A, linos AB, AC parallel to PH, QK, cutting HK at B, C. K Then ABC is the required triangle. Proof. L BAH /. AHP, since AB is parallel to PH. Z.BHA= Z.AHP, constr. BH = BA. Similarly it may be proved that CK = CA. /. AB -f BC + CA = HB + BC + CK - HK. Also Z.ABC= /.PHK^ Z.X, corresp. /.s. and Z.ACB= Z.QKH - /.Y, corresp. /.s. .". ABC is the required triangle. Q.E.F. CONSTKUCTION 9 Draw a triangle given one angle, the side opposite that angle and the sum of the other two sides. B > B/*' .' A ; P A^ r C O Fio. 284. 18 274 CONCISE GEOMETRY Given two lines a, p and an angle X. To Construct a triangle ABC such that BC = a, BA + AC =j Draw a line PQ and from it cut off a part PC equal to p. Construct a line PB such that L BPC equals -J L X. With C as centre and radius equal to a, draw an arc of a circle, cutting PB at B. Construct the perpendicular bisector of PB and let it meet PC at A. Join AB, BC. Then ABC is the required triangle Proof. Since A lies on the perpendicular bisector of PB, AP-AB. /. ZAPB=Z.ABP. .'. L BAC - L APB + Z. ABP. = 2ZAPB. = Z X since Z APB -= \ L X. Also AB + AC = AP + AC-PC = />. and BC = a, by construction. .". ABC is the required triangle. Q.K.F. Note. Since there are two possible positions of B, namely, B and B ; , there are two triangles which satisfy the given conditions. CONSTRUCTION 10 Given the angle BAC, construct points P, Q on AB, AC such that PQ is of given length and the angle APQ of given size. A P\ H B Fio. 285. Given the angle BAC, a link I and an angle X. To Construct points P, Q*on AB, AC such that PQ equals I and _ APQ equals /.X. CONSTRUCTIONS FOR BOOK I 275 Take any point H on AB and construct a line HK such that Z.AHK= ^.X. From HK cut off HF equal to I. Through F draw FQ parallel to AB to cut AC in Q. Through Q draw QP parallel to FH to cut AB in P. Then PQ is the required line. Proof. By construction, PQFH is a parallelogram, Q.E.F. and ZQPA = ^FHA= Z.X, .*. PQ is the required line. CONSTRUCTION 11 Describe a square on a given straight line. -t- Q ** A FIG. 286. Given a line AB. To Construct a square on AB. From A draw a line AC perpendicular to AB ; from AC cut off AP equal to AB. Through P draw PQ parallel to AB. Through B draw BQ parallel to AP, cutting PQ at Q. Then ABQP is the required square. Proof. By construction, ABQP is a parallelogram. But L BAP = 90, /. ABQP is a rectangle. But AB = AP, .*. ABQP is a square. Q.K.F. CONSTRUCTIONS FOR BOOK II CONSTRUCTION 12 (1) Reduce a quadrilateral to a triangle of equal area. (*J) Ueducu any given rectilineal figure to a triangle of equal area. D JC ( 1 )\Given a quadrilateral ABCD. To Construct a triangle equal in area to it. Join BD. Through C, draw CK parallel to DB to meet AB produced at K. Join DK. Thou ADK is the required triangle. Proof. The triangles BCD, BKD are on the same base BD and between the same parallels BD, KC. .". area of ABCD = area of ABKD. Add to each AABD. .". area of quad. ABCD area of A AKD. /. AKD is the required triangle. D C Q.E.F. FIG. 288. 276 FOR BOOK IT 277 (2) Given a pentagon ABODE. To Construct a triangle equal in area to it. Proceed as in (1). This reduces the pentagon to a quadri- lateral AKDE of equal area. Proceed as in (1). Join EK; through D draw DL parallel to EK to meet AK produced in L. Then AEL is the required triangle. This process can be repeated any number of times. C< >NSTIU ICTTON 1 3 Bisect a triangle by a line through a given point in one sid<\ A 0, ' H B PC FIG. 289. Given a point P on the side BC of the triangle ABC. To Construct a line PQ bisecting the triangle. Supi>ose P is nearer to C than B. Bisect AB at H. Join PH. Through C, draw CQ parallel to PH to meet AB at Q. Join PQ. Then PQ is the required line. Proof. Join CH. Since AH = HB, area of AAHC - area of ABHC. .'. ABHC = JAABC. Since HP is parallel to QC. Area of AHPQ = area of AHPC. Add to each, ABHP. .*. area of A BPQ = area of ABHC. But ABHC = |AABC. /. ABPQ=JAABC. .'. PQ bisects A ABC. Q.B.F. 278 CONCISE GEOMETRY If it is required to draw a line PQ, cutting off from the triangle ABC a triangle BPQ equal to a given fraction, say |, of the triangle ABC, take a point H on BA such that BH = |BA and proceed as in the above Construction. CONSTRUCTION 14 Divide a given straight line into any given number of equal parts. h'' c ,KX \ F- A A P Q R 3 B FIG. 290. Given a line AB. To Construct points dividing AB into any number (say 5) equal parts. Through A, draw any line AC. Along AC, step out with compasses equal lengths, the number of such lengths being the required number of equal parts (in this case 5). Let the equal lengths be AF, FG, GH, HK, KL. Join LB, and through F, G, H, K draw lines parallel to BL, meeting AB at P, Q, R, S. Then AP, PQ, QR, RS, SB are the required equal parts. Proof. Since the parallel lines FP, GQ, HR, KS, LB cut off equal intercepts on AC, they cut off equal intercepts on AB. Q.E.F. CONSTRUCTION 15 Divide a quadrilateral into any number of equal parts by lines through one vertex. Given a quadrilateral ABCD, To Construct lines through A which divide ABCD into any number (say 5) equal parts. Join AC; through D draw DP parallel to AC to meet BC produced at P. CONSTRUCTIONS FOR BOOK II 279 Divide BP into the required number (in this case 5) of equal parts, BQ V QjQ 2 , Q 2 Q ? , Q H Q 4 , Q 4 P. Through those points which lie on BC produced, in this case Q 3 Q 4 , draw lines Q 3 R 3 , Q 4 R 4 parallel to PD to meet CD in R 3 , R Then AQ^ AQ 2 , AR 3 , AR 4 are the required lines. Proof. By construction, the AABP and the quad. ABCD are equal in area. But the areas of AS BAQ^ QjAQ,,, Q 2 AQ 3 , Q 3 AQ 4 , Q 4 AP are equal, fc* their bases are equal and they have the same height. /. each - 1 A ABP = I quad. ABCD. Further, AACQ 3 =AACR 3 , AACQ 4 = AACR 4 , AACP = AACD, being on the same base and between the same parallels. /. AAR 3 R 4 =r AACR 4 - AACR 3 = AACQ 4 - AAQ 3 Q 4 . And similarly AAR 4 D = AAQ 4 P. Also quad. AQ 2 CR 3 = AAQ 2 C + AACR a . AACQ 3 . .*. AQ 1? AQ 2 , AR 3 , AR 4 divide quad. ABCD into five equal parts. x Q.E.F. Note. The same method may be used for dividing a rectilineal figure with any number of sides into any number of equal parts, either bylines through a vertex or by Jines through a given point on one of the sides. CONSTRUCTIONS FOR BOOK III CONSTRUCTION 16 Construct the centre of a circle, an arc of which Ls given. Q A Km. 292. Given an arc AB of a circle. To Construct the centre of the circle. Take three points P, Q, R on the arc. Construct the perpendicular bisectors OX, OY of PQ, QR, intersecting at O. Then O is the required centre. Proof. The perpendicular bisector of a chord of a circle passes through the centre of the circle. .*. the centre of the circle lies on OX and on OY, * the centre is at O. Q.E.F, CONSTRUCTION 17 Construct a circle to pass through three given points, which do not He on a straight line. *--. j*.- Fio. 293. 880 CONSTRUCTIONS FOR BOOK III 281 Given three poin ,,, _, _. To Construct a circle to pass through A, B, C. Construct the perpendicular bisectors OX, OY of AB, BC, intersecting at O. With O as centre and OA as radius, describe a circle. This is the required circle. Proof. Since O lies on the perp. bisector of AB, OA = OB. Since O lies on the perp. bisector of BC, OB = 00. .". the circle, centre O, radius OA, passes through B, C. Q.E.F. CONSTRUCTION 18 (1) Construct the inscribed circle of a given triangle. (2) Construct an escribed circle of a given triangle. A FIG. 294. Given a triangle ABO. To Construct (1) the circle inscribed in A ABC. (2) the circle which touches AB produced, AC produced and-CC. (1) Construct the lines Bl, Cl, bisecting the angles ABC, ACB and intersecting at I. Draw IX perpendicular toBC. With I as centre and IX as radius, describe - 282 CONCISE GEOMETRY This circle touches BC, CA, AB. Proof. Since I lies on the bisector of /. ABC, I is equidistant from the lines BA, BC. Since I lies on the bisector of Z.ACB. I is equidistant from the lines CB, CA. I is equidistant from AB, BC, CA. the circle, centre I, radius IX, touches AB, BC, CA. (2) Produce AB, AC to H, K. Construct the lines Bl r Cl^ bisecting the angles HBC, KCB and intersecting at I A . Draw I 1 X 1 perpendicular to BC. With \ l as centre and \ 1 X 1 as radius, describe a circle. This circle touches AB produced, AC produced and BC. Proof. Since \ L lies on the bisector of /_ HBC, l t is equidistant from BH and BC. Since \ l lies on the bisector of /.KCB, Ij is equidistant from CK and CB \ l is equidistant from HB, BC, CK. the circle, centre I 1? radius \ l X i , touches HB, BC, CK. Q.E.F. CONST RUCTION 1 9 (1) Construct a tangent to a circle at a given point on the circum- ference. (2) Construct the tangents to a circle from a given point outside it Fi<4. 295. (1) Given a point A on the circumference of a circle. To Orr*,stritt the tangent ac A to the circle. Construct the centre O of the circle. Join AO. CONSTRUCTIONS FOR BOOK III 283 Through A, construct a line AT perpendicular to AO. Then AT is the required tangent. Proof. The tangent is perp. to the radius through the point of contact. But AO is a radius and /_ OAT = 90, .". AT is the tangent at A. (2) Given a point T outside a circle. To Construct the tangents from T to the circle. Construct the centre O of the circle. Join OT and bisect it at F. With centre F and radius FT, describe a circle and let it cut the given circle at P, Q. Join TP, TQ. Then TP, TQ are the required tangents. Proof. Since TF=FO, the circle, centre F, radius FT, passes through O, and TO is a diameter. .*. L TPO = 90 = Z. TQO. L in semicircle. But OP, OQ are radii of the given circle. TP, TQ are tangents to the given circle. Q.E.F. CONSTRUCTION 20 (1) Draw the direct (or exterior) common tangents to two circles. (2) Draw the transverse (or interior) common tangents to two non-intersecting circles. 284 CONCISE GEOMETRY (1) Given two circles, centres A, B. To Construct their direct common tangents. Let a, b be the radii of the circles, centres A, B, and suppose a>6. With A as centre and a b as radius, describe a circle and construct the tangents BP, BP' from B to this circle. Join AP, AP' and produce them to meet the circle, radius a, in Q, Q'. Through Q, Q' draw lines QR, Q'R' parallel to PB, P'B. Then QR, Q'R' are the required common tangents. Proof. Draw BR, BR' parallel to AQ, AQ' to meet QR, Q'R' at R, R'. ]}y Construction, PQRB is a parallelogram. .'. BR=PQ = AQ- AP-a -(-&)=&. R lies on the circle, centre B, radius b. Also, since BP is a tangent, /. BPA = 90. /. L RQA = 90 and L. BRQ - 90, by parallels. .'. QR is a tangent at Q and R to the two circles. Similarly it may be proved that Q'R' is also a common tangent. (2) Given. Two non-intersecting circles, centres A, B. To Construct the transverse common tangents. Let a, b be the radii of the circles, centres A, B. With A as centre and. a -f b as radius, describe a circle and construct the tangents BP, BP' to it from B. Join AP, AP', cutting the circle radius a at Q, Q'. Through Q, Q' draw lines QR, Q'R' parallel to PB, Then QR, Q'R' are the required common tangents. CONSTRUCTIONS FOR BOOK III 286 Proof. Through B draw BR, BR' parallel to AQ, AQ' to meet QR, Q'R' at R, R'. By construction, PBRQ is a parallelogram. .*. R lies on the circle, centre B, radius 6. Also, since BP is a tangent, /_BPA=90. /. *- AQR = 90 and /. BRQ = 90, by parallels. .". QR is a tangent at Q and R to the two circles. Similarly it may be proved that Q' R' is also a common tangent. Q.B.F. CONSTRUCTION 21 On a given straight line, construct a segment of a circle containing an angle equal to a given angle. AT B \ \ c\ FIG. 299. fiven a straight line AB and an angle X. To Construct on AB a segment of a circle containing an angle equal to Z.X. At A, make an angle BAG equal to /. X. Draw AD perpendicular to AC. Draw the perpendicular bisector of AB and let it meet AD at O. With O as centre and OA as radius, describe a circle. Then the segment of this circle on the side of AB opposite to C is the required segment. Proof. Since O lies on the perpendicular bisector of AB, OA = OB; /. the circle passes through B. Since AC is perpendicular to the fadius OA, AC is a tangent ; /. /.X== /.CAB = angle in alternate segment.' Q.E.F. 286 CONCISE GEOMETRY CONSTRUCTION 22 Inscribe in a given circle a triangle equiangular to a given triangle. FIG. 300. Given a circle and a triangle ABC. To Construct a triangle inscribed in the circle and equi- angular to ABC. Take any point R on the circle and construct the tangent XRY at R to the circle. Draw chords RP, RQ so that ^PRY= Z.CBA and Z.QRX = Z.CAB. Join PQ. Then PQR is the required triangle. Proof. L PQR = L PRY, alt. segment. and Z.QPR= Z.QRX, alt. segment. = Z.CAB. .'. the remaining Z.QRP = the remaining Q.K.F. CONSTRUCTION 23 Describe about a given circle a triangle equiangular to a given triangle. /Y P FIG. 301. CONSTRUCTIONS FOR BOOK III 287 (liven a circle and a triangle ABC. To Construct a triangle with its sides touching the circle and equiangular to ABC. Construct the centre O of the circle : draw any radius OP and produce PO to Q. Draw radii OR, OS so that ^QOR = Z.ACB and Z.QOS = Z.ABC. Draw the tangents at P, R, S, forming the triangle XYZ. Then XYZ is the required triangle. Proof. Z.ORZ = 90= Z.OPZ since PZ, RZ are tangents. ORZP is a cyclic quadrilateral. L QOR = /. PZR, ext. L cyclic quad. = int. opp. Z. But Z.QOR= /.ACB, constr, Z.PZR-Z.ACB. Similarly Z.PYS-ZABC. .'. the remaining Z.YXZ of the AXYZ^the roinaiiiing CONSTRUCTION 24 Construct a circle to pass through a given point A and to touch a given circle at a given point B. FIG. 302. Construct the centre O of the given circle. Construct the perpendicular bisector of AB and produce it to cut OB, or OB produced at P. With P as centre and PB as radius, describe a circle. This is the required circle. Proof. Since P lies on the perpendicular bisector of AB, PA=PB. Since P lies on OB, or OB produced, the two circles touch at B. Q.B.F. 288 CONCISE GEOMETRY CONSTBUCTION 25 Construct a circle to touch a given circle and to touch a given line ABC at a given point B on it. B C 'D FIG. 303. Construct the centre O of the given circle. Construct the perpendicular BD to AB and cut oft* a part BE equal to the radius of the given circle. Construct the perpendicular bisector of OE and produce it to cut EB, or EB produced at P. With P as centre and PB as radius, describe a circle. This is the required circle. [There are two solutions according to which side E is of AC.] Proof. Let PO cut the given circle at Q. P lies on the perpendicular bisector of OE. /. PO=PE. In Fig. 303, BE=OQ and PE=PO. /. PB = PQ. Also P lies on OQ produced. /. the circle, centre P, radius PB, passes through Q and touches the given circle at Q. Q.E.F. CONSTRUCTIONS FOR BOOK IV CONSTRUCTION 26 Divide a given finite straight line in a given ratio (i) internally, (ii) externally. p C ' - A X B FIG. 304. Given two lines p, <? and a finite line AB. AY v\ To Construct (i) a point X in AB such that _=-. XB q (ii) a point Y in AB produced such that =-?. x ' eBY ? (i) Draw any line AC and cut off successively AH=jt>, HK = <. Join KB. Through H draw a line parallel to KB to cut AB at X. c, B Y - FIG. 305. (ii) Draw any line AC ; cut off AH =^>, and from HA cut off HK = . Join KB. Through H draw a line parallel to KB to cut AB produced at Y? 19 290 CONCISE GEOMETRY CONSTBUCTION 27 Construct a fourth proportional to three given lines. ,'Y j > _ O P Q X FIG. 306. Given three lines of lengths a, b, c units. To Construct a line of length d units, such that - = c . o a Draw any two lines OX, OY. From OX cut off parts OP, OQ such that OP = a, OQ = b. From OY cut off a part OR such that OR = c. Join PR. Through Q, draw a line QS parallel to PR to meet OY at S Then OS is the required fourth proportional. Proof* Since PR is parallel to QS OP OR OQ~OS' Q.E.F. Note. To construct a third proportional to two given lines, lengths a, b units, is the same as constructing a fourth proportional to three lines of length a, 6, b units. CONSTRUCTION 28 To construct a polygon similar to a given polygon and such that corresponding sides are in a given ratio. Given a polygon QABCD \md a ratio XY : XZ. To Cwutruot a polygon OA'B'O'D' such that 9*' = OA A'3' . XY * t= * , = AB XZ CONSTRUCTIONS FOR BOOK IV 291 Join OB, OC. Draw any line OQ and cut off parts OP', OP equal to XY, XZ. Join PA. Y Z Through P' draw P'A' parallel to PA to meet OA at A'. Through A' draw A'B' parallel to AB to meet OB at B'. Throiigh B' draw B'C' parallel to BC to meet OO at C'. Through C' draw C'D' parallel to CD to meet OD at D'. Then OA'B'C'D' is the required polygon. Proof. Since A'B' is parallel to AB, AS OA'B', OAB are similar, OA' = A'B' __ OB' *" OA ~ AB OB' . ., , OB' B'C' OC' , Similarly = = -, and BO on. OA' = A'B' _ && ^ Qjy = D'O " OA AB ~~ BC """CD"" DO" A1 OA' OP' XY Also OA--6P--55- /. the sides of OA'B'C'D' are proportional to the sides of OABCD in the ratio XY : XZ. Further, by parallels, the polygons are equiangular. /. the polygons are similar and tjjieir corresponding sides are in the given ratio. Q.E.F. 292 CONCISE GEOMETRY CONSTRUCTION 29 Inscribe a square in a given triangle. ,'X' Y'' V... _J X Y FIG. 308. Given a triangle ABC. To Construct a square with one side on BC and its other corners on AB and AC. On BC describe the square BXYC. Join AX, AY, cutting BC at X', Y'. Through X', Y' draw X' B', Y'C' parallel to XB (or YC) to cut AB, AC at B', C'. Join B'C'. Then B'X'Y'C' is the required square. x, , v , AB' B'X' AX' X'Y' AY' Proof. By parallels AB BX AX XY AY Y'C' = AC' YC " AC .'. Since A = , B'C' is parallel to BC and ^^= fA'. AB AC BC AB .'. B'X'Y'C' is similar to BXYC and is .". a square. Q.E.F. The following is a more general but less neat method. A A > B' / \ 0' t *'\ t \ / \ t *"--* *"*-. Q F FIG. 309. ' Q CONSTRUCTIONS FOR BOOK IV 293 Take any square PQRS with PQ parallel to BC, and circum- scribe a triangle FGH about this square equiangular to ABC. [Draw SF, RF parallel to AB, AC ; produce FS, FR to meet PQ produced at G, H.] Divide BC at X' in the ratio GP : PH. Then X' is one corner of the square ; complete by parallels and perpendiculars. CONSTRUCTION 30 Construct a mean proportional to two given lines. Given two lines of lengths a, b units. i To Construct a line of length x units such that - = - or x* = ab Of FIG. 310. METHOD L Take a point O on a line and cut off from the line on opposite sides of O, parts OA, OB of lengths &, b units. On AB as diameter, describe a circle. Draw OP perpendicular to AB to cut the circle at P. Then OP is the required mean proportional. Proof. Produce PO to meet the circle at Q. PQ is a dhord perpendicular to the diameter AB, /. PO = OQ. But PO . OQ = AO . OB, intersecting chords of a circle. or = OP~ b ' Q.E.F. 294 CONCISE GEOMETRY MKTHOD II. Take a point O on a lino and cut off from the line on the name side of O, parts OA, OB of lengths a, b units. Q On OB as diameter, describe a circle. Draw AQ perpendicular to OB to meet the circle at Q. Join OQ. Then OQ is the required mean proportional. Proof. /. OQB = 90 ; angle in semicircle. .". OQ is a tangent to the circle on QB as diameter. But Z.QAB = 90, /. circle on QB as diameter passes through A. OQ 2 = OA . OB, tangent property of circle. OQ.-... or Q.B.P. Note. In practical constructions, Method II. is often pre ferable to Method I. CONSTRUCTION 31 (i) Construct a square equal in area to a given rectangle, (ii) Construct a square equal in area to a given polygon. FIG. 312. (i) Given a rectangle ABOD. To Coitstruct a square of equal area. Produce AB to E, making BE = BC. CONSTRUCTIONS FOR BOOK IV 295 On AE as diameter, describe a semicircle. Produce CB to meet the semicircle at P. On BP describe a square. This is the required square. Proof. By the proof of Constr. 30, BP 2 = AB. BE, but BE=BC. .'. BP 2 = AB . BC = area of ABCD. QE.F. (ii) Given any polygon. To Construct a square of equal area. Y Q K Z FIG. 313. By the method of Constr. 12, reduce the polygon to an equivalent triangle XYZ. Draw the altitude XK and bisect YZ at Q. Use (1) to construct a square of area equal to a rectangle whose sides are equal to YQ and XK. This is the square required. Proof. Area of polygon = area of A XYZ. = JYZ . XK. = YQ.XK = square. Q.E.F. CONSTRUCTION 32 (i) Construct a triangle equal in area to one given triangle and similar to another given triangle. (ii) Construct a polygon equal in area to one given polygon and similar to another given polygon! (i) Given two A* ABO, PQR. To Construct a AXBZ equal to AABC and similar to Suppose APQR placed with QR parallel to BC. 296 CONCISE GEOMETRY Through A draw a line AD parallel to BC. Through B draw BH parallel to QP to meet AD at H. Through H draw HK parallel to PR to meet BC at K. C Z K Q R Fm. 314. Construct the mean proportional BZ to BC, BK. Through Z draw ZX parallel to KH to meet BH at X. Then XBZ is the required triangle. Proof. By parallels, AXBZ is similar to AHBK and /. to APQR. BC . BK BC AABC AHBK BK 2 BK 2 BK .'. A XBZ = A ABC. Q.E.F. (ii) Given two polygons F and OSTUV. To Construct a polygon OS'T'U'V similar to OSTUV and equal to F. JJ' FIG. 315. Reduce the two polygons F and OSTUV to equivalent triangles ABC, PQR respectively and proceed as in (i). [See Fig. 314.] c On OS take a point S' such that OS = . OS QR On O& construct the polygon* OS'T'UV similar to OSTUV. Then OS'T'U'V' is the polygon required. Proof. CONSTRUCTIONS FOR BOOK IV ^9 S'T'U'V' os' 2 ez 2 AXBZ AABC OSTUV * os 2 QR 2 APQR APQR F OSTUV' .*. OS'T'U'V'=F. Q.E.F. Note the use made of Theorems 58, 59. CONSTRUCTION 33 Construct a circle to pass through two given points and touch a given line. C P 'O Q D FIG. 316. Given two points A, B and a line CD. To Construct a circle to pass through A, B and touch CD. Join AB and produce it to meet CD at O. Construct the mean proportional OG to OA, OB, and cut off from CD on each side of O parts OP, OQ equal to OG. Construct the circles through A, B, P and A, B, Q. These are the required circles. Proof. Since OA . OB = OG 2 = OP 2 = OQ 2 , OP, OQ are tangents to the circles ABP, ABQ. Q.E.F. Note that the method fails if AB is parallel to CD. This special case forms an easy exercise. CONSTRUCTION 34 Construct a circle to pass throbgh two given p6int ahd touch a given circle. 298 CONCISE GEOMETRY Given two points A, B and a circle S. To Construct a circle to pass through A, B and touch S. Construct any circle to pass through A, B to cut S at C, D say FIG. 317. Produce AB, CD to meet at O. From O, draw the tangerfts OP, OQ to S. C6nstruct the circles through A, B, P and A, B, Q. These are the required circles. Proof. OA . OB = OC . OD, property of intersecting chords. = OP 2 = OQ 2 , tangent property. .". OP, OQ are tangents to the circles A, B, P and A, B, Q, .". these circles also touch S. Q.E.F. CONSTRUCTION 35 Construct a circle to pass through a given point and touch two given lines. B FIG. 318. CONSTRUCTIONS FOR BOOK IV 299 Given two lines AB, AC and a point D. To Construct a circle to touch AB, AC and pass through D. [The centres of all circles touching AB, AC lie on a bisector of L BAG.] Draw any circle touching AB, AC and let P bo its centre ; P being in the same angle BAC as D* Join AD and let it cut the circle at Q, Q'. Draw DE parallel to QP to meet AP at E. With centre E and radius ED, describe a circle. This circle will touch AB, AC. Proof. If EH, PX are the perpendiculars from E, P to AB. circle, centre E, radius ED, touches AB at H. Similarly it may be proved to touch AC. A second circle is obtained by drawing DE' parallel to Q'P to meet AP at E'. Q.K.P. ANOTHER MKTHOD. Take the image of D in the bisector of L BAC, call it D'. By the method of Constr. 33, draw a circle to pass through D, D' and to touch AB ; this circle will then touch AC. CONSTRUCTION 36 Construct a circle to touch two given lines and a given circle. B' Given two lines AB, AC and a circle S, centre* D, tacfius r. To Construct a circle to touch AB, AC, and S. 300 CONCISE GEOMETRY Draw two lines A'B', A'C' parallel to AB, AC and at a dis- tance r from them. By Constr. 35, draw a circle to touch A'B', A'C' and to pass through D. Let O be its centre. With O as centre, draw a circle to touch AB. This circle will also touch AC and S. Proof. Let P', Q' be the points of contact with A'B', A'C'. Let OP', OQ', OD cut AB, AC, S at P, Q, E. Then PP' = QQ' = r = ED ; but OP' = OQ' - OD. .'. OP=OQ=OE and OP, OQ are perp. to AB, AC. .". the circle, centre O, radius OP, touches AB, AC, S. Note. There are in all four solutions : this construction gives two solutions, since two circles can be drawn to touch A'B', A'C' and pass through D. And by drawing A'B', A'C' at dis- tance r from AB, AC on the other side, two other solutions are obtained. CONSTRUCTION 37 Bisect a triangle by a line parallel to one side. Given a triangle ABC. To Construct a line parallel to BC, cutting AB, AC at P, Q so that PQ bisects AABC. Bisect AB at F. Construct the mean proportional AG between AF, AB. From AB cut off AP ecjual to AG. Draw PQ parallel to BC, cutting AC at Q. Then PQ is the required Kne. Proof. AABC AB 2 AB 2 AB CONSTRUCTIONS FOR BOOK IV 301 CONSTRUCTION 38 Divide a given line into two parts so that the rectangle contained by the whole and one part is equal to the square oti the other part. C A X B FIG. 321, Given a line AB. To Construct a point X on AB so that AB . BX = AX 2 . Draw BC perpendicular to AB and equal to |AB. Join CA. From CA cut off CP equal to CB. From AB cut off AX equal to AP. Then X is the required point. Proof. Let AB = 21. BC = I BX = 21- and AX 2 = Z 2 ( ^5 - 1) 2 = J 2 (6 - 2 .'. AB.BX = AX 2 . CONSTRUCTION 39 Construct an isosceles triangle, given one side and such that each base angle is double of the vertical angle. A 302 CONCISE GEOMETRY Given a side AB. To Construct a triangle ABC such that AB = AC and /.ABC With centre A and radius AB describe a circle. On AB construct a point P such that AB . BP = AP 2 . Place a chord BC in the circle such that BC = AP. Join AC. Then ABC is the required triangle Proof. AB . BP = AP 2 , but AP - BC. .'. AB.BP-BC 2 . BC touches the circle APC. .'. ZBCP=ZCAP. AS BCP, BAC are equiangular [ Z ABC is common | But AB = AC, /. CB = CP. ButCB = AP, /. CP = PA. .'. ZPAC-ZPCA. But Z PAC = Z PCB, /. Z BCA - 2 /. PAC or 2 Z BAC. .'. Z ABC - Z BCA - 2 Z BAC. A -- Note. FIG. 323. Since the angles of a triangle add up to 180. Z.ABC Z BCA =72 and Z BAC = 36. /. BC is the side of a regular decagon inscribed in the circle. From C, draw CH perpendicular to AB and produce it to meet the circle at D ; then CH = HD and Z. CAD -- 72. CD is the side of* a regular pentagon inscribed in the circle. The following result is useful :-y- If p and d are the lengths of the sides of a regular pentagon CONSTRUCTIONS FOR BOOK IV 303 and a regular decagon inscribed in a circle of radiua a, In Fig. 323, let AB = a, CD=*p, CB = d; it is required to prove that p 2 = a 2 -f d 2 . Since AB. BP=*BC 2 and BP=BA - AP=BA- BC = a-d. /. a(a-d) = d 2 ora 2 -ad-d 2 = o. From ACHB, CH 2 + HB 2 = CB 2 ; but CH - JCD = \p and /. /> 2 = a 2 + d 2 - 2(a 2 - ad - d*). .". jt? 2 = a 2 4- ^ 2 , since a 2 - ad ~ d 2 = o. CONSTRUCTION 40 Inscribe (i) a regular pentagon ; (ii) a regular decagon in a given circle. F FIG. 324. Let A be the centre and EAB a diameter of the given circle. Let AF be a radius perpendicular to AB. Bisect AE at G. With G as centre and GF as radius, describe a circle, cutting AB at P ; join PF. Then AP and PF are equal in length to the sides of a regular decagon and a regular pentagoji inscribed in the circle. The regular figures are therefore constructed by placing chords in the circle end to end equal to these lines. Proof. From GF cut off GR equal to GA ; froip FA cut off FS equal to FR. 304 CONCISE GEOMETRY Then by Constr. 38, FA . AS= FS 2 . NowGR = GAandGP = QF, .'. AP=RF- = or. But AF = AB, .*. BPAS. .'. BA. BP = AP 2 . /. by Constr. 39, AP is equal to a side of the regular decagon. But AP 2 + AF 2 =PF 2 . /. PF is equal to a side of the regular pentagon. (See pp. 302, 303.) 306 CONCISE GEOMETKY NOTES 307 308 CONCISE GEOMETRY NOTES 809 310 CONCISE GEOMETRY NOTES 311 312 CONCISE GEOMETRY GLOSSARY AND INDEX ACUTE angle : any angle less than 90. Alternate angle, 5. Altitude : the altitude of a triangle is the perpendicular from any vertex to the opposite side. Angle in a semicircle ; an angle whose vertex lies on the circumfer- ence and whose arms pass through the extremities of a diameter. Apollonius* theorem, 226. Arc of a circle : any part of the cir- cumference. Area of circle, 86. Area of triangle and trapezium, 27. Bisect : divide into two equal parts. Centroid, 98. Chord : the line joining any two points on the circumference of a circle. Circle : the locus of a point which is at a constant distance (called the radius) from a fixed point (called the centre) is called the circumfer- ence of a circle. Circumcentre, 97. Common tangents, 283 Complementary angles : angles whose sum is 90. Concentric : having the same centre. Congruent : equal^ in all respects. The symbol is zz. Corresponding angles, 5. Cyclic quadrilateral : a quadrilateral whose four corners lie on a circle. Decagon : a figure with ten sides. Degree : $Vth part of a right angle. Depression, angle of, 145. Diagonal : the line joining two op- posite corners of a quadrilateral. Diameter : a chord of a circle passing through tho centre. Elevation, angle of, 145. Fquilateral : having all its sides equal. Equivalent : equal in area. Exeentre, 97. External bisector: if BAC is an angle and if BA is produced to X, the line bisecting Z. CAX is called the external bisector of BAC. Hexagon : a figure with six sides. Horizontal line : a line ]>erpen<1wular to a vertical line. Hypotenuse : the side of a right- angled triangle opposite the right angle. Identities, geometrical, 228. Image, 93. Incentre, 97. Isosceles triangle : a triangle with two sides equal. Locus, 248. Mean proportional, 121. Median : the line joining a vertex of a triangle to tho mid-point of the opposite side. Mensuration formulae, 86. Nine point circle, 102. Obtuse angle : an angle greater than 90* and less than 180. Octagon : a figure with eight sides. Orthocentre, 98. 318 Parallel lines, 208 Parallelogram, 22. 314 GLOSSARY AND INDEX Pedal triangle, 98. Pentagon : a figure with five sides. Perimeter: the sum of the lengths of the sides bounding a figure. Perpendicular : at right angles to. Playfair's axiom, 208. Projection, 224. Proportional (third or fourth), 290. Pythagoras' theorem, 222. Rectangle, 22. Reflection, 93. Reflex angle . an angle greater than 180. Regular polygon : a polygon having all its sides and all its angles equal. Rhombus, 22. Right angle, 205. Sector of a circle : the area bounded by two radii of a circle and the arc they cut off. Segment of a circle : the area bounded by a chord of a circle and the arc it cuts off ; a segment greater than a semicircle is called a major seg- ment, if less a minor segment. Similar, 257. Square, 22. Supplementary angles : angles whose sum is 180. Symbols : = equal in area. == congruent. ~ the difference between X and Y is repre- sented by X ~ Y. > greater than. < less than. Z angle. A triangle. || umtn parallelogram. O ( ' circumference. Tangent, 243. Trapezium, 22. Vertical line : a line which when produced passes through the centre of the earth. ANSWERS. 1. Where only one form of unit occurs in the question, the nature of the unit is omitted in the Answer. 2. Answers are not given when intermediate work is unnecessary. 3. Results are usually given correct to three figures, and for angles to the nearest quarter of a degree. EXERCISE I (p. 2) 5. 6; 11; 22. 7. 135. 8. 83; 112i ; 167. 9. (iv) 300 ; (v) 990. 10. 20. 12. (ii) 65. 13. 120. 15. 120. 16. 72. 17. 72. 18. 120. 19. 2474. 20. 5*. 24. 40. 25. 110 ; 149J . 26.15. 27.46. 28.111. 29. 111J. 80. 251. 81. 180 -IK. 32. EXERCISE III (p. 10) 5. 122. 6. 93. 7. 80. 10. 36. 12. 80. 13. 80. 14. Least is 36. 15. 8. 16. 37. 17. 86. 19. 2a;-180 . 20. 120. 21. |(g - y) + 90. 25. 162. 27. y = ~, y=6, 10, 18, 42. 28. 6. 31. x=c-a-b. 8-a? 32. XU & C* 33. x EXERCISE IV (p. 16) 3. (i) 90, 45 ; (ii) 72, 36. 6. 60, 60, 70. 6. 3=360-2?. 7. a=60Jy. 9. 36.* 33. 26f. EXEROJSE.V (p. 23) 5. 68. 7. 62. 23. 67i. 315 316 CONCISE GEOMETRY EXERCISE VI (p. 28) 1. 7'5. 2. 17-5 ' f 3. 4-8, , 4. 4. 5. 42. 6. 44. 9. 4-8. 10. 12. 11. 675. 12. 10*5. 13 375. 14. 4-5; 4. 15. 4-8. 16. 15. 17. 4-8; 4-8. 18 4-4. 19. 26. 20. 8. 21. 6'2; 20. 22. 4' to milo ; I". 23. b(zq + xr + yp + yq\ 34. Uvr + (rr-\-Qs). 25. W. r 26. 24; 12 ; 36. 28. (*</ -<?/) 29. 5; 10. 30. (i)4 ; (ii)- r >; (in) 5T> ; (iv) \<*c ; (v) ^(iid - be). 31. (i)10 ; (ii) 11. 32. (i) 3- 3; (ii)6-4. 33. (i) 14- 7, fi-88; (ii)57'2 ,H 3. 34. 5-56. EXERCISE VII (p. 38) 1. 13 . 2 . 8. 3. 5-66. 4. 32-25. 6. 9 A. 6. 5-83. 7. 217. 8 . 477. 10. 30. 11. 14970. 12. 17-3; 1 975 ft. 13. 21 1. 14. 16-2 mi. 15. 60 yd. 16. 4-47* 17. 5 , 18. 5. 19. 6-93. 20 . 2-89. 21. 5. 22. 5 ; 7. 23. 13. 26. 6., 27. 88 28. 55-2. 29. 5-46. 30. 3-57. 31. 7 . 33 9- 16. 34. 8-66. 35. 26' 8. 36. 18. 37. 6 24. 39. Each side 60 sq. in. ; 117 'in. 40. 7'34. EXERCISE VIII (p. 44) 1. (0, (), (iv), 2. 19. 3. If ; 2-67. 4 . 5-85 ; 6-84. 5. 11; 1; 6-93. 6. 42-43. 7. 6-63. 8 . 12-2, 10. Ves. 13. 3 5. 14. 5'45 ; 6-52 ; 7-97. 15. 9-17. 16. 10. 17. 127. EXERCISE X (p. 49) 13. 7. EXERCISE XI (p. 52) 21. 12* ; 17*. EXERCISE XII (p. 57) 1. 9-16. 2. 13.' 3. 11-5. 4. 7*V 5. 8'58, 0*58. 6. 5-38. 7. 3-46. 8. 5. 9. 4. 10. 8. 11. 4-8. 12. 3*12. 13. a; 2 + xy = a? -Jr. 14. 5'22. EXERCISE XIII (p. 62) 1. 40. 2. 55", 3. 110. 4. 37. 5. 107. 6. 100; 110 7. 54 ; 99. 8. 105. 9. 72. 10. 124. 11. 54. 12. 105. SWEIIS :*> 1 7 XIV (p. 68) 1. 62. 2. 117 W . 3. 2t>, 8. 1 58, 64. 5. 103", 90, 77, 90. 6. 94, 8. 7. 120. KXEKCISE XV (p. 72) 1. 30, 45, 105 or 15, 30, 135. 2 74, 22J, 150 or 22 J u , 30 \ 1274". 4. 3:1. 5. 46, 37. EXERCISE XVI (p. 77) 1. 3. 2. 2'5, 1-5, 4'6. 3. 8, 4, 3. 4. 5 '3, 3 '6, 4 '5. 5. 10-5, 1-6. 6. 6 7. 1J. 8. 32, 8. 9. 3. 10. 1-5, 2-5. 11. '5, 2-5. 12, 12. 13. 19-1, 12. 14. 7, 1. 15. 4-45, 11-125. 16. 5 -3\/2~-0'757'. 18.1-44,36. 19.24. 20.1+^2 = 2-41. EXERCISE XVIII (p. 87) 1. 25-1 in., 50-3 sq. in. ; 628 yd., 31,420 sq. yd. 2. 0'8. 3. 1-1. 4. 2-1. 5. 5-89. 6. 4'57. 7. 57 18'. 8. 3'2. 9. 158-5. 11. 84-8. 12. 21-5. 18. 628; 408. 14. 3. 15. 25. IS, 314 ; 204. 17. jj. 18. 288, 19. 48 ; 96 20. 65'4; 78'5. 21. 100,000, 000 sq. m. ; 4. 22, 8'2. 23. 9'21. 24. 20'1. 25, 2f. 26. 78-5. 27. 514; 500; 9'0. 28. 119; 44-0. 29. 77-4. 30. 828'5 sq. ft, 34. 11'8. 35. 29'3. 36. 102'5. 37. 8 ; 14 ; 1, 38. 6'86; 137; 186. EXERCISE XIX (p. 94) 32. 20 in. EXERCISK XXI (p. 100) 2. (iv)l I- 5. 1 i ; or 1, 7. 6. 8. t'g- 10. 3'2. 11. 6. 15. 2 : 5 ; 1:2 16. 1-6" 18. 3-2 21. x~: y . 22 2 ^ ; y . x ~~ y 9i i i ia__Jfc'(/>- ^~'j ~t - 7* + ^) 25. 1 i '2(3 + *"* .; J - y*' x+y g+r A- 1' 27. *i- 28. 1-6, 29. * M-&A 41. 1. EXERCISE XXII (p. 112) 1. 120. 2. 4ft. 4. 10 5 x8'6 mi. ; 10* x 2 '3 mi. 5. 6' 8 ;/ . 6. 66. 7. 14-4". 8. 6'4, 7 '2 cms. 9. 224. 10. 1-5, 8{. 11. 5. 12. 8i 13. (i) S, ? ; (ii) 64 ; (iii) 2f, 11 ; (iv) 54 ;8. 15. 2*4. 16. 18, 8. 17. 7'2. 18. 14 r 19. 34, 11. 20. 12-8, 5. 21. 8f. 22. 4. j 23. (1) 2J;(ii) 7*4 5y = 35. 24. 2-9. 25. 12. 2$. If 27. 6, ll. 28. 3J. 39. (i)54 ', 24' , (ii) 13". 30. 3A. , 3*. y* S* 82. y=-> , ) ^ J % ** / 84, t/ ~ J *x as. luVf Ar> * 318 CONCISE GEOMETRY EXERCISE XXIII (p. 122) 2. 6. 3. I j. 4. 10. 5. 2 or 10. 6. (i) 6 ; (ii) 12 ; (iii) 2-31 ; (iv) 21^. 7. 4, 6J. 8. H. 11. 7-07 ; 13*04. 12. 0707. 13. J?5l ^?!L gi-^j*-^ EXERCISE XXIV (p. 127) 1. 12 sq. ft. 2. 40. 0. 9. 6. 101J. 8. 4'2. 9. 3*75 sq. in. 10. 16 : 4 : 3 ; 9. 12. 5i. 13. 4if. 15. 512. 16. 1*024. 17. 6. 18. 2s. 3d. 19. 9if. 21. 40 -6 ; 162. EXERCISE XXV (p. 132) 1. 3, 15. 2. 3-35. 4. 12. 5. 9J. 6. 3 sq. in. EXERCISE XXVI (p. 135) 30. 4*8. 69. 81 45' or 14 40'. EXERCISE XXVII (p. 145) 1. 94-3. 2. 7140. 3. 13' 9". 4. 10' 8". 5. 32. 6. 2-77. 7. S. 37 W. ; 5'17 mi. 8. 7*0 mi. N. 34 W. 9. 8-42 mi. N. 12 W. 10. E. 86| N. 11. 34 -8 mi. N. 31J W. 12. 10-6. 13. 321. 14. 91'9, 15. 85'3. 16. 2'59. 17. 84'0. 18. 326. 19. 31. 20. E. 59 S. 21. 177. 22. 137. 23. 34 '4. EXERCISE XXVIII (p. 148) 3. 3-36. 6. 2-5. 7. 6'13. 8. 2'83. EXERCISE XXIX (p. 150) 1. (i) 36 50' ; (iii) 2'59 ; (iv) 2'93 ; (v) 479 ; (vii) 6'68 ; (viii) 5'66, 3'53 ; (xii) 11-3 ; (xiii) 8'49 ; (xiv) 8*87 ; (xv) 1044. 8. 574. 9. 5 '23. 10. 10tfi. 11. 49i- 12. 62J. 13. 5'41. 14. 2*55. 15. 7*13, 3'63. 16. 49J. 17. (i) 4-96 ; (ii) 676 ; (iii) 5'18 ; (iv) 63J ; (v) 3'82. 18. (i) 25i ; (ii) 8'25 ; (iii) 6 ; (iv) 6 -21. 19. 8 -64. 20. 3 '53. 21. 4*67. 22. 7. 23. 6'09. 24. 6-16. 25. 4*26. 26. 4*96. 27. 4-62. 28. (i) 7'67; (ii) 7'10; (iii) 10*1 ; (iv) 478; (v) 7'82; (vi) 871. (vii) 6 -64. 29. 6-22. 30. 5 '34. EXERCISE XXX (p. 153) 10. 1*63. 11. 21|. EXERCISE XXXI (p. 155) 1. (i) 10; (ii) 50-0; (iii) 147; (iv) 6; (v) 48; (vi) 9'43 ; (vii) 45-1; (viii) 28 ; (ix) ?l t (X) 18. 2. 15*0.- 8. 575. 4. 4*57. 5. 30. 6. 2'64. 7. 36j. 8. 40. 9. 4'07. 10. 5*80 or W6. 13. 29'1 ANSWERS 319 EXERCISE XXXII (p. 168) 6. 1-93. 7. 3-61. 8. 6*82". 9. ffl j. . 10. 3'1 7, EXERCISE XXXIII (p. 161) 5. 6-65. 17. 0-64, 1'16, 1-93, 5 '80. 18. 1-4(5. 24. 2'13. 26. 3'11. 27. 1-94. 28. 4'61. EXERCISE XXXIV (p. 172) 6. 4'47. 9. 320. 14. 2'66. 15. 1-66. 16. 5 -80. 17. 1'32. 18. 8-13. 24, 5'60, 2'14. 25. 6'Ob*, 4 '02. 29. 5'87, 2*23. 30. 6-89, 4'89. 35. 4'16. 37. 11J. EXERCISE XXXV (p. 175) 1. 7*5. 2. 7'2. 7. (i)2-89; (ii) 10'3. 11. 4'12 ; 1-21. 16. 3'63. 20. 2*27. 21. 4-55. 22. 2*68. 23. 5'36. EXERCISE XXXVI (p. 178) 1.6-325. 3.6-08. 4. 7 '36 or - 1 '36. 5. 5 '29. 6.3*29. 7. 5-00. 19. a =7 '22 or -2*22, y=2'22 or -7'22. EXERCISE XXXVII (p. 180) 3. 10. 4. 578. 5. 4-81. 7. 3'83. REVISION PAPERS (p. 181) 1. 300. 6. 112. 10. 110. 29. 75. 33. a = 540-a-&-c. 37. 67J. 41. z=180-a-&-a;-y. 42. *? rt. angles. 46. 80. n 49. 375. 53. ^(xy-{-yz). 66. 4*24. 57. 18'4(5). ' 61. 2J. 64, 5-5, 2-5, 17'3. 65. 4[p(y + r) + q(r + s) + x(s~y)]. 68. 13". 69. 15, 9. 72. 9. 73. 300. 77. 7*5. 78. 2-16. 81. 12; 5'66. 85. 2. 88. (ii)>/* T ^8iT416 ; (iii)a;>6J. 89. 2. 99. 47. 102. 60, 80. 106. a * + * h \ lh 107. 55, 40. 109. 13. 115. 16. 118. E. 25 N. 9V 131. 17. 133. on AB 10, on CD 20. 186. 43'2. 137. 6. 144. ~. 147. f. 155. 9i, J. 159. 6J in. 167. 6, 10, 14 in. 170. 132. 174. 4'47. 175. 24 -4 in. 177. 6, 5. 179. 2*2. 182. 0-69 or 23'3. 183. 5| f 2J T . 187. 2. 189. 3-2, 1*2, 4-4. 194. 687. \95. 2, 2?. 197. 4. 199. 4-8. I'KINTKD BY M OK IU SON AND GIHH LIMIT* KDINKUROH